Inequalities Involving Lower-Triangular Matrices

INEQUALITIES INVOLVING LOWER-TRIANGULAR MATRICES By P. D. JOHNSON J B and R. N. MOHAPATRA [Received 26 November 1976—Revised 25 July 1979] Introduction Throughout the paper the scalars will be real or complex numbers. The space of all sequences of scalars will be denoted by co, and the subspace of co consisting of sequences with only finitely many non-zero entries will be denoted by <p. Spaces lp, c0 will have their usual meaning. || • ||j, will denote the usual norm, if 1 ^ p ^ oo, or quasinorm, if 0 < p < 1, in lp. For x = {xn} G co, y = {yTC} e co, xy is the sequence {#n?/n}. If x e co, and A 9 to, then x\-{xy\y e A}. If x e co, then |*| = {|aJn|}> and if a; is a non-negative sequence and p a real number, then xp = {xnv}. If xn ^ 0 for each n then ar 1 = {av 1 }. If, as will sometimes be the case, d is a sequence whose entries are non-zero scalars or oo, then dr1 = {d^1} has zero entry where d has entry oo. For A = {amn)mtn>1 an infinite matrix of scalars, the domain of A, denoted by dom-4, is given by dom^4 = \x G ID: 2 a>mnxn *s a convergent series for each m), and for x G dom^l, Ax = {(Ax)^ is given by 00 \Ax)m = 2 amnxw We shall denote by / the identity matrix and by A1 the transpose of the matrix A. Throughout this paper, A = (amn)m>n^i will be a lower-triangular matrix (i.e. amn = 0, for m < n) with non-negative entries and positive entries on the main diagonal (i.e. amm > 0, for m = 1,2,...). The lower triangularity implies dom^4 = co. The assumption of non-zero diagonal entries implies that A has a two-sided matricial inverse, denoted by A~x. A~x is also lower triangular, although not with non-negative entries, unless A is diagonal. Since the map x -> A~xx of co onto co is the inverse of the map x -> Ax, A~x is a transformational as well as a matricial inverse of A. In an attempt to give a simple proof of Hilbert's inequality [10, Theorem 315], Hardy [8] established the following; which later went by the name 'Hardy's inequality'. Proe. London Math. Soc. (3) 41 (1980) 83-137 84 P . D. JOHNSON J B AND R. N . MOHAPATRA THEOREM A. If p > 1 and un ^ 0 {for n = 1,2,...), (i) ti n=l\ fc=l / n=l XP-L/ unless all the un's are zero, or u = {wn} £ Zp. The constant at the right of (1) was fixed by Landau [14] who showed that it is the best possible for each p. Since then many alternative proofs of Hardy's inequality have been given (cf. Broadbent [1], Elliot [5], Grandjot [7], Knopp [13], Kaluza and Szego [12]). Copson [2] generalized Theorem A by replacing the arithmetic mean of sequence u by a weighted arithmetic mean. We shall consider Copson's generalization in § 10. If A = {amn) is the Cesaro matrix amn = rnr1, where m ^ n, then Hardy's inequality can be written as (2) for 1 < p < oo. Petersen [18] and subsequently Davies and Petersen [4] produced sufficient conditions on a matrix A and an auxiliary sequence/ £ co for the existence of an inequality of the form \\A\x\\\p^K\\f.a.x\\p (l^p<co) for some K with a = {amm}, the main diagonal sequence of A. Our object in this paper is to study inequalities of the form or, indeed, of the form (3) M i z i u , with (A, || • ||A), (/x, || • 11^) being quasinormed FK spaces satisfying certain hypotheses. Henceforward, inequalities of the form (3) will* be referred to as HPD inequalities (for Hardy, Petersen, and Davies). Particular attention will be paid to the existence of best possible HPD inequalities, for fixed A, (A, ||*||A), and (/x, ||11^); best possible not by virtue of the smallness of the number K, but of the sequence b, with respect to a particular definition of 'smallness' of sequences (see § 3). I t is clear that the smaller the K, the better is the inequality (3). Just so, the 'smaller' the 6, the better is the inequality (3). We shall obtain conditions on A} an (^>II'IIA)> d 0*ilHI/») for *n© existence of a 'smallest' b for which (3) holds for some K and will apply our results with A,/u among the lr and A specified or confined to some special class of matrices. LOWER-TRIANGULAR MATRICES 85 In our investigations we shall not concern ourselves with the determination of the smallest possible K even in specific cases. In order to obtain the best possible HPD inequalities we shall study some multiplier spaces of sequences and develop machinery which we can use to decide whether or not an HPD inequality is possible in a specific case. It might be stated that there is no best possible HPD inequality for Theorem A. (See § 9 for the proof.) Inequalities of the form (3) amount to an inclusion relation between two sequence spaces; in §§2 and 3 we discuss the appropriate inclusions. 1. Definitions and background Let N denote the positive integers. For 8 £ N, let e^ be the sequence with 1 at each entry corresponding to an integer in 8, zero elsewhere. For short, if S = {n}, then es = en, and if 8 = N, then es = e = {1,1,...}. If x, y E a) and the series 2n=i xnyn converges, then the number to which the series converges will be denoted by (x,yy. If x, y are nonnegative sequences, then (x,y)> will still denote S^Li^nS/n ©v©n if this series diverges. The nth coordinate projection on w is the linear functional < •, en>. For each n e N, Po is the zero map of a» into w. A subspace A of CJ with a topology T is called an FK space provided the following hold: (i) (A, T) is a topological vector space; (ii) (A, T) is a complete metric space; and (iii) < •, en} is continuous on (A, T) for each n e N. Note that we do not require FK spaces to be locally convex, as some authors do. We shall use the following facts about FK spaces which are consequences of the Banach-Steinhaus and closed graph theorems: (i) if one FK space is included in another then the inclusion map is continuous; (ii) an FK space is an FK space with one and only one topology; (iii) a matrix map from one FK space into another is continuous [24, p. 204, Corollary 5]. It is clear that (i) and (ii) can be derived from (iii) by taking the matrix / . If x G co, the normal hull of x is N{x) = {yew: \yn\ ^ \xn\ for n e N). 86 P . D. JOHNSON J B AND R. N. MOHAPATRA A subset A of o> is normal if x e A implies N(x) c A. If A <= to, a subset fi of A is said to be normal in A if and only if x e ft implies N(x)nX <= p. Clearly, if A is normal, normal in A means just normal, for subsets of A. If A is a subspace of o>, a normal topology on A is a t.v.s. topology on A for which there exists a neighbourhood base at the origin consisting of sets which are normal in A. If A is a normal subspace of co, and T is a normal topology on A, then < •, en> is continuous on (A, T) for all n e N if and only if (A, T) is a Hausdorff space (see [11, Proposition 2.7]). Thus a normal subspace of oo with a complete metric, normal topology is an FK space. If (E, T) is a t.v.s., a function p from E into the non-negative reals will be said to determine the topology T if the sets {x e E: p(x) < e}, with e > 0, comprise a neighbourhood base at the origin in {E, T). If A is a subspace of oo, a function p from A into the non-negative reals will be called normal if and only if a; 6 A and y e XnN(x) imply p(y) ^ p(x). If the normal function p on A determines a topology T on A, then T is a normal topology. The topology of a Hausdorff t.v.s. is determined by some function p if and only if the space is metrizable, in which case p may be replaced, if desired, by a pseudonorm [20, Theorem 6.1, p. 28]. In this paper the topology-determining functions will be norms or quasinorms, although some of the more general considerations are applicable to spaces with more disturbing topology-determining functions. (We are thinking of the spaces l{pn), inequalities related to which will not be studied here; see [15] and [22].) A quasinorm || • || on a vector space!? is like a norm except that, instead of the triangle inequality, || • || satisfies || x+y \\ < M(\\ x \\ +1| y \\) for all x,y e E, for some M > 1. {li M = 1,||-|| is a norm.) If (^,||-b), (F, || • H^) are quasinormed spaces, a linear map T from E to F is continuous if and only if ||T|| = supy^jlira;!^ < oo; ||-|| is a quasinorm on S?(E,F), the space of continuous linear transformations from E to F, and is a norm if | | - b i s . It might be remarked that for 0 < p < l,||«||p is a quasinorm on lp, with M = 2 1 / P- 1 . For A ^ o>, we define (4) nor-^-^A) = {x: A \ x \ e A}. It is known (see [11, Proposition 1.3]), under conditions on A less stringent than those in force here, that if A is a normal subspace of to, then so is nor-J4~1(A), and furthermore that nor-J.-1(A) is the largest normal subset of -4-1(A) (thence the notation nor-^4.~1(A)). Let S(A, A) = {n e N: the nth column of A is in A}. LOWER-TRIANGULAR MATRICES 87 Then if A is a normal subspace of co, it is easily deducible from the preceding remarks t h a t 8(A,X) = {neN:ene = N\{n e N: xn = 0 for sAlxe Suppose A is a normal subspace of co with normal topology T, and % is a neighbourhood base at the origin in (A, T) consisting of normal sets. I t is shown in [11, Theorem 2.3] that the sets HOT-A-\U) = {xe nor-^-^A): A | x \ e U}, for U e W, comprise a neighbourhood base at the origin for a normal topology nor-^4~1(T) on nor-^4~1(A). If T is determined by the normal function p, then nor-^4~1(T) is determined by jo, defined by p so defined is also normal. If p is a (quasi) norm, then so is p. Suppose A is a subspace of co invariant under Pn for each n e N (as will be the case, for instance, if A is normal), and T is a t.v.s. topology on A. Then (A, T) is said to be AK (short for Abschnitt-Konvergenz) if and only if Pn{x) converges to x in (A, T), as n -> oo, for each x e X. In the sequel, we shall depend upon the following facts either contained in the foregoing summary or easily deducible therefrom. If A is a normal subspace of co and T is a normal topology on A, then (i) if (A, T) is FK and AK, then so is (nor-^-^A (ii) if A = Za, and T is the usual norm topology, then isFK; (iii) if T is determined by the normal (quasi) norm || • ||, then nor-4~1(2T) is determined by the normal (quasi) norm \\A |'|||. Regarding the completeness of (nor--4~1(A), nor-^4~1(2T)) we have the following. PROPOSITION 1.1. Let Xbe a normal subspace of co with normal topology T and let (A, T) be a complete Hausdorff space. Then (nor-^4"1(A), nov-A-x(T)) is complete. Proof. Let °U be a normal base of neighbourhoods of the origin in (A, T). Let {x{n)} be a Cauchy sequence in (nor-J4-1(A),nor-^4-1(T)), where xln) denotes the sequence {x^}. Thus we are given that, given any U 6 % there is an n0 such that, for n,m^ n0, (5) 88 P . D. JOHNSON J B AND B . N. MOHAPATRA Now, using the suffix j to denote the jth coordinate, we have \(Ax{n)-Ax{m))j\ V an x(v(ri) _ x/v.(m) \ ZJ jk\ k k I = k=l j Zi^fcl^A; x k I and a V a jk\ \xk I I xk \) jk\xk Thus, since ?7 is normal, (5) implies that Ax<n)-Ax(m) e U, Thus {-4a;(n)}, {^4|rc(n)|} are both Cauchy sequences in (A,T); since (X,T) is complete they must converge. Their limits must be a transform of some sequences. Thus there are sequences x,y (say), with Ax, Ay e A such that (6) ^la:(n) -> ^a:, (7) ^l|a;(w)|-^^- Now by the facts that A, T are normal and (A, T) is a Hausdorff space, the coordinate functions are continuous (see [11, Proposition 2.5]). Since, this is also for given k, (^l"1^)^. is a linear combination of xvx2,...,xk, continuous; hence it follows from (6) and (7) that cr (n) and k as n -> oo. Hence y = \x\. Thus A \x\ = Ay e A, so that x e nor-u4~1(A). Also we can write (7) as (8) A \x n I -> A \x\. Although we do not quite want this, after observing that x e nor-^4~1(A), we can go through the above argument again with x{n) replaced by LOWER-TRIANGULAR MATRICES {n) x -x; 89 and (8) will then yield the desired result, A\x{n)-x\-+0 as n ->• oo. n o r - ^ l " 1 ^ ) is complete, with the norm REMAEK. 2. For jit, A c o>, D(fi, A) = {x e to: x/x c A}. Here are the essential facts about D(ix,X) to be borne in mind through what follows. (a) If A is a subspace of co, then so is D({JL,X). (b) If either /u. or A is normal, then D(/x, A) is normal. (c) If A is a subspace of to, then D^, A) is the largest normal subset of A. Thus, if A is a normal subspace of to, then (d) If \x is normal and A is a subspace of to, then D{\i, A) = D(fi, £)(£«» A)); the inclusion => follows from the fact t h a t D^oo, A) <= A, and the inclusion <= follows from the observation that, for any x e co, Xfx. is normal if fi is, so if XfM c= A, then X/M c D ^ ^ A ) , since this last is, according to (c), t h e largest normal subset of A. I t follows t h a t if \i is normal, and A is a normal subspace of o>, then D(fi,A~1(X)) = D(fj,, nor-A~1(X)). For A c to, D(X, Ij) is called the Kothe-Toeplitz dual of A and is denoted by Ax. (Ax)x is denoted by Xxx, for short. Note t h a t x e Xx if and only if < I x U V I) < oo for each y e A. If A = Xxx, X is said to be perfect. Xx itself is perfect, for any subset A of co. PROPOSITION 2.1. For any /x,A c oo,D(/x,A) c D(Xx,nx), with equality if X is perfect. Proof. Suppose 6 e D(n, A), x e Xx, and 2/ e /u,. Then <|teUy|> = <l*l,|ty|><«> since by e A. Thus bx e nx. Thus D(fj,,X) g D(Xx,fxx). Now suppose A is perfect. Then D{Xx,nx) c D ^ , Axx) = D(^XX,A) £ D(/u,A), since /u £ ^«B. ThusD(Ax,/Ltx) = i)(^,A). For a; e co, let Z>z denote the diagonal matrix with diagonal sequence x. Note t h a t Dx(y) = rcy for each y e co. If/x and A are subspaces of co, the association of b e D({x, A) to the linear transformation x -> bx from /A t o A is itself a linear transformation of D((JL, A) into the space of all linear maps from fi t o A. This association is an injection if (j, satisfies the following: (9) 'for each n e N there exists x e /a such t h a t xn ^ 0*. 90 P. D. JOHNSON JB AND R. N. MOHAPATRA In the case where /x satisfies (9), D(/z, A) will be identified with its copy in the space of linear transformations from /u, to A. If fi and A are FK spaces and fi satisfies (9), then JD(/X, A) c SB{\i, A), for the action of b e D(fx,X) on [A is the same as that of the matrix Db, and a matrix transformation between FK spaces is continuous. PROPOSITION 2.2. Suppose (/x, || • 1^), (A, || • ||A) are quasinormed FK spaces, and [ju satisfies (9). Then D(fx,,\) <= J?((x,X) is, with the operator quasinorm || '||, an FK space. If ||*||A is normal, or if both p and ||*|| are normal, then 11*11, restricted to D([M,X), is normal. Proof. Let p denote the dual norm on A', the space of continuous linear functional on A; that is, p(f) = sup|yA4S1|/(y)| for each f E A'. Suppose n E N. By (9), there exists x e fx such that xn # 0. Suppose b e D([JL,X). Then | <&, en> | = | xn~\bx, en) | < | s n -i | P « •, e n » || bx ||A Thus the coordinate projections are continuous on (D(/u,,A),||*||). Now, suppose (b{n)) is a Cauchy sequence in (D(fx,\),||*||). Because (j^(/Lt,A),||*||) is complete, (bM) converges to some TGJS?(/X,A), with respect to ||*||. Because the coordinate projections are continuous on (D{[ji, A), 11*11), (b{n)) converges coordinate-wise to some 6 e o>. For each x e fi, (b{n)x) converges to Tx in (A, || • ||A); since the coordinate projections are continuous on (A, || • ||A), and bln) -> b coordinate-wise, it follows that bx = Tx. Thus b e D{fx, A), and (bM) converges to b in (£(/*, A), ||• ||). Thus (Z)(/u,,A),||*||) is complete, and is therefore an FK space. Now suppose fj. and ||*||^ are normal, b E D(fj.,X), and c E N(b). Then c = bd for some d e l^ with \\d\\m ^ 1. If x E /H, and 11*11^, < 1, then dx E N(x) c ^ and \\dx\\^ ^ Hall^ < 1. Thus ||e||= sup \\bdz\\x* sup ||&z||A Now suppose || • ||A is normal, b e D(n, A), and c E iV(6) n D(/x, A). For each x e fji, ex e XnN(bx), whence ||ca*||A ^ ||6^||A. Therefore, ||c||= sup \\cx\\x^ sup ||6a;||A = l l | | l I l l l l REMARK. If \i and A are FK spaces but not necessarily quasinormed, it may still be the case under certain conditions, for instance that p be locally bounded, that D(/x,A) is an FK space when equipped with the relative topology of uniform convergence on bounded subsets of /x. However, we shall not use any conclusion of this sort, more general than that of Proposition 2.2, in this paper. LOWER-TRIANGULAR MATRICES 91 From Proposition 2.2 and the remarks of §1, we have the following corollary. COROLLARY 2.3. Suppose (ytx, ||*||A) and (A, ||-||A) are quasinormed FK spaces, and suppose X and ||*||A are normal. Suppose /z satisfies (9). Then D(fj,, nor-A~1(X)) is a normal FK space, with the normal quasinorm \\-\\ defined by \\b\\ = s u p m ^ \ \ A \ b x \ | | A . Suppose (A, ||-||A) and (/x, || *||^) satisfy the hypothesis of Corollary 2.3, and b e D(fx,nor-A^iX)). Then, since ||-4|'||| A is the quasinorm on (10) l for all x G /x, with K = || b ||, the operator quasinorm of b. It is inequalities of the form (10) that we will, in fact, study; they will also be called HPD inequalities, although (10) is manifestly not the same as (3). The two forms are interchangeable when \J, satisfies a minor (for the spaces we are going to consider) additional hypothesis. 2.4. Suppose (A, ||«||A) and (/a, || • ||A) satisfy the hypothesis of Corollary 2.3, and, in addition, fi is invariant under multiplication by es, for each S ^ N. Suppose b e co. Let S = {n: bn ^ 0}, and define hn~x = bn~x for n G S, h^1 = 0 otherwise. (a) / / b e D{fi,nor-A-1^)), then \\A\x\\\x^ KWt^xW^ for all xebp, with K = || 61|, ||* || being the operator quasinorm. and H*^ is normal, then (b) / / || ^41 a; | ||A < A" || 6a; 1^ for all xeh'1^, PROPOSITION Proof, (a) Suppose x = by, y e fx. Then x = b.es.y, by the definition of S, and esy G p. Thus , = \\A\b.es.y\\\, r (b) It is implicit that 6"1 G D(ix,noT-A~\X)). Suppose x e /z. Then a;|||A < K||66-ix||, = since ||*||A is normal and esx G N(x)nfx. For fixed A, (A, || • ||A), and (/x, || • || ) satisfying the hypothesis of Corollary 2.3, HPD inequalities (10) arise from, and only from, elements 6 of D()Lt,nor-^4-1(A)). Thus to study all HPD inequalities (10) is to study D(/x, nor-^4~1(A)), and that sums up the plan of this paper. 92 P . D. JOHNSON J B AND R. N. MOHAPATRA 2.5. Suppose (A, ||-||A) and (/*, || • l y satisfy the hypothesis of Corollary 2.3 and, in addition, fx and || • || are normal. Then the map b -* ADb is a quasinorm-preserving linear map of D(fi, nor-A^X)) into £?(fi,X). PROPOSITION Proof. Clearly b -> ADb is a linear map of D([JL, nor-A~1(X)) into some set of matrices. Now, if b e D(fM, nor-A~1(X)), then bfx, = Db(p) e nor-A-^X) c A^X), so A(Db(ix)) = {ADb) ([M) <= A. This inference relies on the facts that A is the transformational inverse of A'1, and that for MVM2 any lowertriangular matrices, and x e to, Mx(Mz{x)) = (M-^M^) (x). Thus ADb maps /x into A, if b e Z)(/u,,nor-u4~1(A)), and so represents a continuous map of \i into A, since a matrix transformation between FK spaces is continuous. The operator quasinorm of ADb e S^(fx, X) is sup MADb){x)\\k= \\ADJ= H sup \\A(bx)\\,. 1 1 1 1 1 For x G n,\\x\\p ^ 1, let, for each n e N,yn be a scalar such that \yn\ = 1 and bnynxn = |6na;tt|. Then yx e ii, since /x is normal, and \\yx\\p = H*!^, since H'H^ is normal. Thus ||-4-D&|| > sup nxi^i || -41 bx | ||A. On the other hand, since the entries of ^4 are positive, the ordinary triangle inequality for numbers shows that A (bx) e N(A\bx\) for any x e to; since ||-||A is normal, it follows that \\ADb\\4 sup H I I I Thus || i!2>61| = sup HJBII ^11-416*11^, which is precisely the operator quasinorm of 6 6 D(/x,nor--4-1(A)). 3. An ordering of a> For a, b e w, we define a < b if and only if a = be for some c el^. If a < b and b < a, a and 6 are called equivalent. Equivalence thus defined is an equivalence relation on to, and < induces a partial ordering of the equivalence classes. However, we shall avoid mentioning these equivalence classes in what follows. Thus, if A c o> and a e to, we say a is a maximum sequence in A (with respect to < ) if it is the case that 6 6 A implies b < a, and do not essay the more proper but cumbersome reference to the equivalence class of a and the partial order induced by < . If a e A c to, a is maximal in A if b e A and a < b imply a and b are equivalent. Note that x and | x \ are equivalent, for each x e CD, and that if A is a normal subspace of w, a, b e a>, and a < b, then aX <= bX. Furthermore, a < b if and only if al^ c blx. LOWER-TRIANGULAR MATRICES 93 The order < provides the notion of 'largeness 'or 'smallness' of sequences mentioned in the introduction. Read a < b as b is larger than, or no smaller than, a. This terminology is a bit more useful than the traditional 0( •) notation, in that it permits us to deal with sequences that may have zero entries. Note that if bn # 0 for all n, then a < b if and only if «• = 0(bn). REMARK. One can take an = O(bn) to mean that there is a constant M such that \an\^Mbn (for all n e N). In this case we can have bn = 0 since this will imply an = 0. But such a meaning attributed to an = 0{bn) will require bn non-negative for all n e N while a < b does not do so. Because of the normality of || • ||A and the non-negativity of the entries of A, inequality (10) is better the larger b is. To ask for a best HPD inequality of the form (10) is to ask for a maximum b e D(fj,, nor--4~1(A)); to ask for a maximally good inequality (10) is to ask for a maximal be PROPOSITION 3.1. Suppose A c w is such that x e X implies \x\ e X and the sum of any two non-negative sequences in X is in X. Then a is maximal in X implies that a is maximum in X. Proof. Suppose a is maximal in A, and b e A. Then | a \ + \ b | e A, and clearly a < | a | +161, so | a | +161 < a, since a is maximal in A. It is also easy to see that b < \a| +1b|. Since ' < ' is transitive, b < \a\ + \b\ < a. We thus see that a is maximum in A since b was arbitrary. COROLLARY 3.2. / / /x c w , and X is a normal subspace of o>, then a maximal element o/X)(/x,nor-J4-1(A)) is a maximum element of Proof. D(/>t,nor-^4~1(A)) is a normal subspace of o», and thus satisfies the hypothesis of Proposition 3.1. PROPOSITION 3.3. If X is a normal subspace of o>, then X possesses a maximum element b if and only if X = bl^. The proof is straightforward. 94 P . D. JOHNSON J B AND R. N. MOHAPATRA REMARKS, (i) The 'only if' conclusion of Proposition 3.3 fails if A is not assumed to be normal. Any proper subspace of ?«, containing e provides an example. (ii) Corollary 3.2 says that the search for maximally good HPD inequalities is the same as the search for best possible HPD inequalities, and Proposition 3.3 says that there is a best possible such inequality for fixed A, (A,|HIA)» and (/x, |HI^)» if and only if D(/Lt>nor--4-1(A)) = bl^ for some b. Theorem 3.5, to follow, will give a necessary and sufficient condition, under certain restrictions on A and /x, for D(/Lt,nor-^L~1(A)) = bl^ for some b e co, and describes the unique (up to equivalence) candidate b for this equality. LEMMA 3.4. Suppose (A, || • ||) is a quasinormed FK space, and both A and ||*|| are normal. Set S = {n e N: en e A}, and define b by bn = HeJI" 1 , for n e S, bn = 0 otherwise. Then X^bl^, and the following are equivalent: (a) A = clm for some c e w; (b) A = W.; (c) 6 G A . Proof. Because A is a normal subspace of cu, n $ S implies xn = 0 for all a; e A. Suppose x E A and n e S. Because || • || is normal, so \xn\<\\x\\\\en\\-i = \\x\\bn. Faints, xeN(\\x\\b)cbln. So As W.. \xn\ = 0 = bn = \\x\\bn. Thus Clearly (b) implies (a). Suppose (a) is true. Then S = {n: cn ^ 0). Now, A is an F K space when equipped with the norm p defined by p(a;) = sup|c n - 1 a; n l» neS for all a; e A. Since an FK space is an FK space under only one topology, p and || • || are equivalent, meaning there exist constants m, M > 0 such that mp{x) ^ || x || ^ Mp{x) for all a; £ A. For n £ S, mP{en) = m|c n -i| ^ | | e j | = bn~i < M\cn-*\t and for n $ 8, cn = bn = 0. Consequently, c and b are equivalent, so A = Woo, and in particular b.e = b e A. If (c) holds, then, since A is a normal subspace of co, bl^ c A, and the reverse inclusion has already been established. REMARK. In the preceding lemma, only the inference (a) implies (b) or (c) requires the completeness of (A, ||*||). LOWER-TRIANGULAR MATRICES 95 THEOREM 3.5. Suppose (A,||-||A) and (/t*, || • ||A) are normal, normally quasinormed FK spaces, in eo, and pQ ix. Let 8 = [n e N: Aen G A} and define d = d{A}X,\\'\\x,lx>\\'\\fl) by Then D(fx, nor-A'^X)) c d"1^, and the following are equivalent: (a) DiiXfiior-A^iX)) = cl^for some c e to; (b) Di^nor-A-HX)) = d-Hw; (c) ADd-i maps [x into A. Proof. The proof is an application of the preceding lemma with (A, || • ||) replaced by (D^nor-^-^A)), ||-||), with ||-|| being the operator quasinorm; (^O^/x., nor-^4.—1(A)), [| • ||) satisfies the hypothesis of the lemma by Corollary 2.3 (<p s ^ implies fj. satisfies (9)). As remarked in § 1, 8 = {n:ene nor-^A)} = {n:ene It remains to be shown that the d~l here is the same as the b in the lemma, and that ADd-i mapping /x into A is equivalent to d~x e D{fi,nov-A~\X)). Note that d~x has zeros just where it should. For ne 8, = sup Now, clearly sup,,,.!,^!^! > He.^" 1 , because ||||e n || A - 1 e n || /t = 1. On the other hand, since ||*||A is normal, if H^l^ ^ 1, then so I^KHe.11^- 1 . Thus \\en\\ = \\en\\-i\\Aen\\x = dn, for neS; consequently d'1 plays the same role with respect to (D{fitnoi-A~1(X))t ||*||) as b does with respect to (A, || • ||) in Lemma 3.4. If d*1 G D{ii,noT-A-l(X)), then ADd-i maps [i into A, as noted in Proposition 2.5. On the other hand, if (ADd-i) (fx) £ A, then d~xix c ^^(A), so d'1 G D{fx, A'1^)) = D{fitnoT-A-'1(X)), since fx is normal. COKOLLARY 3.6. Suppose (A, ||«||A) is a normal, normally normed FK space, satisfying the conditions of Theorem 3.5. Set 8 = {n: Aen e A} and 96 P . D. JOHNSON J B AND R. N. MOHAPATRA define dby dn = || Aen ||A if n e S, dn = oo otherwise. If (p, || • ||^) is a normal, normally quasinormed FK space satisfying fx c= lv and HeJI^ = 1 for all neN, then Dfanoi-A-^X)) = d-Hm. Proof. (OIIA = I .. o U if n e S. Since (A,||*||A) is a complete normed space, and {ADd-i(en): n e N] is bounded in (A, ||*||A), it follows that ADd-i maps Zx into A. Since HeJh = 1 for all n e N, it follows from Theorem 3.5 that D(lv nor-A'^X)) = d'H^. Since [x <= ^ and ||e n || /i = 1, for all neN, ADd-i maps /* into A. Now the result follows from Theorem 3.5. COROLLARY 3.7. Suppose (A, || • ||A) satisfies the hypothesis of Corollary 3.6, and d is as in that corollary. Then for 0 < p < l,Z>(^,nor-^4-1(A)) = d-H^. The proof is clear, from Corollary 3.6. COROLLARY 3.8. Suppose (A, || • ||A) and (p, || • ||^) are normal, normally quasinormed FK spaces, <p g fx and ||en||A = 1. Let A-1 = {bnk), b = {bnv}, and c = { K M « . Then D{p,noT-A-i{\)) s fcd.. Proof. Let ^len denote the ?ith column of ^4 and R{n) be the ?ith row of A' . Write .4" 1 = B = {bnk). We have 1 = <R{n),Aen}. But since R™ has all zeros after the ?ith term and Aen has all zeros before the ?ith term, this is the same as 1 = bnnann. The result now follows from Theorem 3.5. 1 REMARK. Let A = lp (1 < p < oo), and let /x, c be as in Corollary 3.8. If b is the sequence of lv> norms of the rows of A'1, then The trouble with Theorem 3.5 is that it is not usually easy to decide when ADd-i maps fi into A; indeed, deciding in the affirmative usually involves proving the inequality || A \ d-xx \ ||A. ^ K || x 1^, the labour in which is not at all diminished by Theorem 3.5. The good side of Theorem 3.5 is that it provides the only possible candidate, modulo equivalence, for a best possible HPD inequality for fixed A, (A, ||-||A), (/*, ||- H^) as in the theorem, namely || A \ d^x \ ||A ^ isTH^H^, with d as in the theorem, and it provides an easy strategy for proving negative results; if you can find x 6 n such that ADd-i(x) $ A, then there is no best possible HPD inequality associated with A, A, /x. LOWER-TRIANGULAR MATRICES 97 In § 5 there will be some more practical, numerical tests (all sufficient, none necessary) for the existence of best possible HPD inequalities, with A,/x among the lp. 4. The task of analysing D(fx, nor-^4~1(A)) turns out to be almost trivial when A = llt apparently because certain 'summation reversing' difficulties are absent. However, the analysis naturally involves some intermediate results that are of interest in themselves, and will be of use later. If x and y are non-negative sequences, and M is a matrix with nonnegative entries, then (Mx,y~) = (x, ilffy). Seeing this amounts to switching the order of summation, a sure procedure when all the numbers being summed are non-negative reals. Note further that for any x, y G CD, because the entries of M are non-negative. 4.1. Suppose Xisa normal subspace of o>, and Bis a matrix obtained from A by replacing each column of A not in Xxx by a non-negative sequence in Xxx. Let S = {n e N: Aen e A}. Then nor-^4-1(A) c es{Bt(Xx))x, with equality if A is perfect. PROPOSITION Proof. Recall that for n $ S, xn = 0 for all x e nor-J.~1(A). Since A c Xxx,{n: Aen^ Ben) c N\S. Consequently, A\x\ = B\x\ for all x e nor-^4~1(A); also, x =±= esx for all x e nor-J.~1(A). Since each column of B is in Xxx, Xx Q. dom £*. Suppose y e Ax, and xenoT-A-\X). Then < | * | , | # y | > ^ <|aj|,5*|y|> = < 5 | * | , | y | > = <il|*|,|y|> < oo, because A \x\ e A. Thus x e (^(A*))*, so esx = x e e s (5 t (A x )) x . Now suppose A is perfect, and x e es{Bt(Xx))x. Then xn = 0 for n £ S, so B\x\ = -4|a;|. Note that x e (Bt(Xx))x, since this latter space is normal. Suppose y e Ax. Then \y\ e Ax, so x e {Bt(Xx))x implies oo > <\x\,B*\y\> = <B\x\,\y\} = <A\x\,\y\>, which implies A \x\ e Xxx = A. Thus x e nor-^4~1(A). 4.2. Suppose X is perfect, and every column of A is in A. Then nor-J4~ (A) is perfect. COROLLARY 1 Proof. By Proposition 4.1, nor-^L~1(A) = (^4t(Ax))x; if /x c a>, then fj.x is perfect. 5388.3.41 G 98 P . D. JOHNSON J B AND R. N. MOHAPATRA 4.3. Suppose Xis a perfect subspace of a>, ||*|| is a quasinorm on X, and (A,||-||) is FK. Set S = \n\ Aen e A}, dn = \\AeJ, for neS, and dn = oo for n $ S. Then d~\ COROLLARY Proof. For any y e Ax, < •, y} is a continuous linear functional on (A, || • ||); for 2n=i Vni' > e n) is a continuous linear functional on (A, || • ||) for each N, since the coordinate projections are continuous on (A, ||«||), and for each x e A, and each N, N N 71=1 n=l 71=1 Thus the continuity of < •, y} follows from the Banach-Steinhaus theorem for complete metric t.v.s. Let || • |C denote the dual norm on the dual of (A, || • ||). For n e S, y e Xx, \(Aen,yy\ ^ MeJIHyll' = djy\\'. Thus, if B is as in Proposition 4.1, ^(A 1 ) <= dl^, where by dl^ we mean {x G <O : for c e £«,, xn = dncn for all n e S}. Therefore, noting that S = {n: dn~x # 0}, we have COROLLARY 4.4. Suppose (A, ||«||) and d are as in Corollary 4.3. Then for any \x g oi,d~^^x c D(fi,nor-A^X)). Proof. d " V = d-Wfalj) = DifiJ-Hj) c D^nor-^-^A)). 4.5. Suppose (A, ||-||) and d are as in Corollary 4.3 and, in addition, ||*|| is normal, and (A, ||*||) satisfies the conditions of Theorem 3.5. Then, forO<p^ l. COROLLARY Proof. Suppose 0 < p < 1. By Corollary 4.4, d~%r = d-H» s D(lp,nor-A-i(X)). On the other hand, D^nor-^h^A)) c d"1^ by Theorem 3.5, since || e j ^ l for all n. REMARK. Corollaries 3.7 and 4.5 have the same conclusion. The difference between them is that in Corollary 3.7, ||*||A is a norm, and in Corollary 4.5, A must be perfect. 4.6. The common conclusion of Corollaries 3.7 and 4.5 does not hold if, in Corollary 3.7, ||-||A is not a norm, or in Corollary 4.5, A is not perfect. Fix p e (0,1], and take any q e (0,#). Let A = Zfl,||*||A = ||-||3Let x be any sequence of positive numbers, and set A = Dx. Then, in this EXAMPLE LOWER-TRIANGULAR MATRICES 1 1 case, d = x, and nor-^4~ (A) = A' ^) D{lp,noT-A-%)) 99 = d~Hq, whence = D(lptd-\) = d-W(lp,lQ) = dr% with r being the positive number such that p^ + r'1 = q~x. LEMMA 4.7. Suppose M = (mkn) is a matrix with non-negative entries, and each row of M is a non-zero lx sequence. Set mk = *Z,n=imkn> m — {mk}k. Then Proof. Since e e £«,, {Ml^Y £ {Jf(e)}x = ra"1^. If x e lv y e ?«,, then 00 = 2 n=l oo / oo Upl^l) E S &=1 \ n = l n _ _ : oo. 1 Thus m- ^ c (Jf^)*. PROPOSITION 4.8. Let d = d(A,l1) be the sequence of lx norms of the columns of A. Then nor-^4~1(Z1) = d'1^. Proof. The proof is straightforward from Lemma 4.7, Proposition 4.1, and the observations that lx is perfect, with ltx = l^, and that {n: the nth column of A is in Zx} = {n: dn~x # 0}. COROLLARY 4.9. Let d be the sequence of lt norms of the columns of A. Then for /x c= o>,Z)(/x, nor--4~1(Z1)) = d^fi?. Proof. D(fx,,noT-A~1(l1)) = D(fx,d~1l1) = d~1D(fjL,l1) = d~1fix. REMARKS, (i) In Corollary 4.9, if we suppose every column of A is in lv and the corollary is applied with /x = lx to obtain a (best possible) HPD inequality, the result turns out to be rather trivial. The inequality obtained is for some K, or oo m m=ln=l / oo \ —l oo \k=n / n=l Changing the order of summation on the left-hand side shows that the 100 P . D. JOHNSON J B AND R. N . MOHAPATRA inequality is actually an equality, with K = 1, for all x. The additional conclusion that this equality is a best possible HPD inequality, i.e. that (&n SfcLn akn)n $ ^» implies that for no K is it the case that for all x, is easy to prove directly. (ii) With M and m as in Lemma 4.7, the conclusion of that lemma is that M(£«,) and ml^ have the same Kothe-Toeplitz dual. It is not necessarily the case that 31(1^) = ml^, however (although M^) c ml^, always, and equality holds when M is diagonal). For an easy example, take M to be a matrix with all rows equal, in which case M{l<n) = sp(e), one-dimensional. Less trivially, even if M is upper or lower triangular, M(lx) need not equal ml^. For an upper-triangular example, set mkn = n~2 for n ^ h. The sequence b defined by ( nr1, n even, 0, n odd, lies in mfc \ilf (£«,). For a lower-triangular example, take the transpose of this same M, and the same sequence b. (iii) Proposition 4.8 says that nor-^4~1(^1) is always a diagonal copy of lv The same is not true of J4~1(Z1). For example, let A be the matrix with entries 1 on the main diagonal, and on the diagonal just below the main, and zeros elsewhere. By Proposition 4.8, nor-J4~1(?1) = lv If 4~1(/1) were a diagonal copy oflv then -4~1(J1) would be normal, so it would be the case that A-Hl,) = nor-A-^lj) = lv But clearly { ( - l)n}n e A-^ 5. Two results of Petersen and Davies and Petersen Throughout this section A = (amn) will be a lower-triangular matrix with amn ^ 0 and amm ^ 0 for m,n e JV. Further, 'a' will denote the sequence amm. With a view to generalizing Theorem A, Petersen [18] proved the following result for any sequence {un}. THEOREM (11) B. Let a matrix B = (bmn) satisfy the following: bmn>0{n^m), (12) (13) (14) bmn = 0(n>m) 0 ^ bnn/bm.n ^K(0^n^m' bmn/bm>n decreases as n increases (m,n = 1,2, ...); < m); (0 ^ n ^ m' ^ m); there exists an f{m){f(m)\co) such that the matrix (cmn) with cmn =f{m)bmn satisfies (12) and (13) with perhaps a different constant K' in (12); LOWER.TRIANGULAR MATRICES 101 1 p (15) the series Em-i & m i(/(^)) " converges; (16) S bml(f(m))i-v ^ Mbkl{f{m)Y-v. m=k Then for p = 1,2,..., and a constant C depending only on p, 00 (17) / TO \ p 00 S | S & » » K l ) <CX m=l \ n = l / {\ujf(m)bmmr. m=l Subsequently Davies and Petersen [4] extended Theorem B to nonintegral values of p and proved the following: THEOREM C. Let B = (bmn) satisfy (11), (12) and the following: (18) there exists anf(m) (/(m) f oo) such that the matrix {cmn) with Cmn=f(m)b satisfies (12); (19) the series 2 {f(k))~p converges; and (20) 2 (./»)-* ^ M(f(k)?-P. m=k Then (17) holds for 1 < p < oo. A matrix satisfying (11) to (14) was called an M matrix and a matrix satisfying (11), (12), and (18) was called an dt matrix. Given a matrix A, and two sequences / and a of positive numbers, let Klt K2 be positive constants. Suppose / C\i \ [VI ] J? I \ I I (YY\ \ft ^» YT J* I \ ^T r\ T I *M ]ft I7 (K ^ ^+ \ ^C W *C *W7 I and (22) a mklakk ^ KaOLn/au (k ^ Wl). If the matrix A and the sequence/ satisfy (21), the pair (A,f) will be said to be MPD. A necessary and sufficient condition that there should be some / for which (21) holds is that A should have the property cif any column of A has zero element below the principal diagonal, then all the elements below that one are also zero'. If (A,f,a.) satisfies (21) and (22) then the triple is called an MP triple. If (21) holds, then (22) must hold for some a; we can just take ocm = l / / m for all m. Hence as far as the matrix A is concerned MDP and MP are equivalent. Let J = (jnk) be such that 0 (k > n). 102 P . D. JOHNSON J B AND R. N. MOHAPATRA Note that if a: is a non-negative sequence, [k=n in a sequence with all entries oo if x $ lv The main object of this section is to examine carefully Theorems B and C and to give a result related to, but not contained in, them. The most important tool in proving Theorems B and C is the following lemma (first proved for integral p, in a less fine form, by Petersen [18]). LEMMA 5.1 (Davies and Petersen [4, Lemma 1]). Suppose 1 < p < oo and zk ^ 0, where k = 1,2, ...,n. Then n ( 2«* \p n I k \p-\ ^*2«* 2O • fc=l \m=l / fc=l / In the case when p = 1 and 2m=i 2 m = 0» we follow the convention 0° = 1. (a) For 1 < p < oo, and zv ...,z n not all zero, the inequality (23) is strict, by the proof in [4]. (b) For fixed p e [1, oo), p is the best possible constant in (23). This is REMARKS, seen by taking zk = 1, k = 1,2, ...,n, and letting n -> oo. (c) For 0 < p < 1, n ^ 2, and zk > 0, with k = 1,2, ...,ny (24) n \p-l I k j»S«* S O fc=l \m=l / n < / \p 22*1 n ( k \p-l <22fc.2 3 j U=l / fc=l \m=l • / The inequality on the left is proved as in [4], and the fact that p is the best possible constant can be seen from an argument similar to that indicated in (b), above. To see the other half of the inequality in (24), set 2**1 A;=i / (2**1 U=i V=i S O/ / verify by differentiation that/(p) increases with p, and note t h a t / ( I ) = 1. The best possible constant in this case is 1, not only for fixed p but also for fixed n. This can be seen by taking zx = 1 and z2, z 3 ,..., zn very small and observing th&tf(p) is close to 1. (d) In view of the inequality (24) one wonders if there can be a companion to the inequality (23) for 1 < p < oo. Indeed, along lines similar to that used in proving (24), one can establish the following. For n ^ 2 and zk > 0, with k = 1, ...,n, n (25) I k \p—\ 5X2M fc=l \ m =l / In \p < 2**1 , U=l / the constant 1 on the right-hand side of (25) being best possible for fixed p and n. LOWER-TRIANGULAR MATRICES 103 Next, we give two theorems which can be proved by modifying the proof of Petersen [18], and Davies and Petersen [4], using Lemma 5.1 and the definitions of MP and MDP. The proofs will be omitted. 5.2. Suppose (A,f,a) is an MP triple, and 1 < p < oo. Then for any x e w, THEOREM (26) || 41 x 11|, < pK^K21| ory*-W{*f^)a \ x\ \\p. (a) The conventions tacitly adopted in the statements above are that O.oo = 0, and that the lp norm of a sequence not in lp, or of any non-negative sequence with oo at some entry, is + oo. (b) Klt K2 used in (26) are the same as those given in the definitions of an MP pair. REMARKS, COROLLARY 5.3. / / (A,f,a) is an MP triple, 1 < p < oo, and for some K3>0, for each n = 1,2,..., then for all x E co. This is the extended version of a theorem of Petersen [18, Theorem 1], to allow non-integral values of p. COROLLARY 5.4. / / {A,f,<x) is an MP triple, and 1 < p < oo, then a./ 1 -^(J fc («/ 1 - p ))- 1 .«- 1 e D(lp,noT-A-l(lp)). THEOREM 5.5. Suppose {A,f) is an MDP pair, and 1 < p < oo. Then for all x 6 co, (27) \\A\x\\\p<pK1P\\fP.J*(f-P).a\x\\\p. This is a special case of Theorem 5.2 in which o^ = l/fm. Since (22) might hold with a 'better' am, Theorem 5.2 may sometimes give a stronger result. COROLLARY 5.6. Suppose {A,f) is an MDP pair, 1 < p < oo, and Then for all x e co, \\A\x\\\p^pK1PL\\f.a.x\\p. COROLLARY 5.7. If (A, f) is an MDP pair, and 1 < p < oo, then 104 P . D . JOHNSON J B AND R. N . MOHAPATRA 1. As noted by Davies and Petersen, Corollaries 5.3 and 5.6 are generalizations of Hardy's inequality (Theorem A). (To see this take A to be the Cesaro matrix; in Corollary 5.3, take fn = n, ocn = l/n, and in Corollary 5.6, take fn = n.) However, these corollaries do n o t quite give the constant p/(p — 1) of Theorem A, even though Kx = Kz= 1, since SfcL n ^~ p *s n o t l e s s * n a n o r equal to (p- l ) - ^ 1 ^ . 2. The 'strictly less than' relation in Hardy's inequality for non-zero sequences of lp (1 < p < oo) is not obtainable from either of the corollaries referred to above. REMARKS. In this paper, however, we shall be concerned with neither strictness of inequalities nor the question of obtaining best possible constants. We next state and prove a theorem along the same lines as Theorem 1 of Petersen [18]. THEOREM 5.8. Suppose, for a positive constant K, that (28) a^ ^ Kakt {t ^ k ^ m), and b is the sequence of lx norms of the columns of A (that is, bn = Sm=n amn)Then for 1 < p < oo, and all u e <o, (29) Proof. If bk = oo, t h a t is, 1/6^ = 0 for jfc = l , 2 , . . . , (29) is trivial. Otherwise, let N be sufficiently large so that l/bk > 0 for some k < N. Suppose un ^ 0 for all n, and un > 0 for some n ^ N. Using Lemma 5.1, we have N ( m \v m-l\fc=l / N Ik TO m=lfc=l iV \p-l \/=l m / \P-1 Ik < pK*-i 2 2amkbk-Hk[ 2««6f S I Nik \p-l Nik <pKv-iX IN U = pRp-i s 2««6rS) A~X S «w \p-l 2a*A-S) «*• Now applying Holder's inequality and dividing both sides of the inequality, so obtained, by / N I TO \P\1/P' ( s (SawArSfc) \TO=1\&=1 / / and making iV^ -» oo, we shall have the required result. LOWER-TRIANGULAR MATRICES REMARK. The conclusion of Theorem 5.8 may be written as 105 \\A\u\\\p<pK*-i\\bM\\p. In fact if all the columns of the matrix A are in llt that is, bn < oo for all n, it is immediate from Theorem 5.8; if, for some n, bn = oo, and if un ^ 0, then the right side of the inequality is infinite so that the result is trivially true. If un = 0, we do not worry about the nth. column of A. Thus it always holds. 5.9. Suppose the matrix A satisfies (28) and b is the sequence of lr norms of the columns of A. Then COROLLARY 6- 1 e D(lp, noT-A-Hlp)) (1 < p < oo). COROLLARY 5.10. Suppose the matrix A satisfies (28), and b and d are respectively the sequences of ^ and lp (1 < p < oo) norms of the columns of A. Then whenever b'1 and d'1 are equivalent, D(lp,noT-A-Hlp)) = d-H*,. Proof. From Theorem 3.5, D(lp,noT-A-Hlp))^d-Hw. 1 Since D(lp, nor-^" ^)) is normal, by Corollary 5.9, The result now follows. (a) Always b"1 < d~x. So the equivalence of b~x and d~x in Corollary 5.10 amounts to d~x < b~x. (b) If all the columns of A are in llt the equivalence of 6"1 and d~x is the same as the equivalence of b and d. In this case, a matrix satisfying the hypotheses of Corollary 5.10 rather resembles a diagonal matrix. (c) By Theorem 3.5, D(lp, nor-A'1^)) £ d'H^ if d~x < x for some x e Dil^noT-A-^lp)) then REMARKS, D(lp,noi-A-Hlp)) = d-Hn. Thus one can obtain corollaries similar to Corollary 5.10 by using Theorem 3.5 and Corollaries 5.4 and 5.7. We shall not state these results, (d) If we set A = (ank) with ank = I/A*"*1 a where An is defined by n=0 {a>0, n,k = 0,1,2,.... n > k), 106 P . D. JOHNSON J B AND R. N . MOHAPATRA then 6, the sequence of lx norms of the columns of A, is given by &=7l = («+!)£ =7l = (a+1) I x^l-xf in view of the integral representation of l/A^ (see [23]). Thus from the remark following Theorem 5.8, we have for 1 < p < oo, (30) £ IW+1)-1 S«*|% (—*)* £ K / 4 , ? (« > o). n=0 L fc=0 J \ a / n=0 However, we do not know if the constant at the right of (30) is' the best possible for fixed p. (Note: the indexing of sequences starts from zero for this example only. Elsewhere in this paper it starts from 1.) The proof of Theorem 5.8 is so apparently a reprise of the proofs of Davies and Petersen's theorems that one might suspect that the result is contained in their results in the sense that the conclusion of Theorem 5.8 might be deducible from either Theorem 5.2 or Theorem 5.5 by an appropriate choice of sequences / and a, or just /. The following example aims at dispelling any such suspicion. I t is important to note that the Corollaries 5.4, 5.7, and 5.9 are equivalent, except for the determination of the constants in the inequalities, to Theorems 5.2, 5.5, and 5.8 respectively. EXAMPLE 5.11. Let the matrix A be such that the nth column of A consists of rn ones, extending consecutively downward from the main diagonal, with all other entries zero. Clearly A satisfies (28), so by Corollary 5.9, (31) r-i e D{lpynoT-A-\lp)) (1 < p < oo). In order to derive this result directly from Corollary 5.7, for any particular p e (l,oo), we would have to find a positive sequence / such that (A,f) is an MDP pair, and r - 1 < f~p(Jt{f~p))~1. Byfixingup the sequence r we shall arrange that for any / such that (A,f) is an MDP pair, f~x $ lp (0 < p < oo) which implies that {Jt{f~p))~1 *s * n e z e r 0 sequence for any p e (0,oo). Thus the result is not deducible from Corollary 5.7. LOWER-TRIANGULAR MATRICES 107 If (A,f) is an MDP pair, then for some K > 0, whence fm-v > K~Pfn-P (n ^ m < n + rn- 1, 0 < p < oo). Let rn > 1 for all n, and define {raj by % = !, ram = m, + r m , - l . Then and, in particular, Thus for m2 < w ^ m3, Continuing this way, we have Hence /w-p ^ /r p i:(^ s fm-p = i s Now, the sequence {raj was formed without reference to / or K. If we arrange for the rmt to be so large that ( diverges for all R > 0 (for instance, take rm = ra!) then (A,f) an MDP pair implies that 2/m~ p = °° f° r aU P G (0, oo)Next, we show that the conclusion (31) cannot be deduced from Corollary 5.4. We shall choose the integers rn in such a manner that if (A,f, a) is an MP triple, then a./1"23 £ lx for all p e (1, oo). If (A,f,a) is an MP triple, then for some positive constants K and L, and all p e (1, oo), we have / m 1 -» > Ki-Pfni-*> (n^m^ n + rn- 1), and Let {mt} be defined as above. Using the fact that (A,f,a) is an MP triple and the definition of {mt}, we have, as before, for 1 < p < oo, 108 P . D. JOHNSON J B AND R. N. MOHAPATRA Once again, a choice of large rm ensures that the last series will diverge no matter how large L, K, and p may be. 6. D(lq)noT-A-Hlp)) We begin with the following: PROPOSITION 6.1. Suppose 0 < p ^ oo, and d is the sequence of lp norms of the columns of A. If for some b G <O, then = d~\lm D(lg,noi-A-%)) for all q e (0,^]. Proof. By Theorem 3.5, if D(lp, nor-A'1^)) D(lp,noi-A-i(lp)) = b.l^, then = d-*ln. For 0 < q < p, lq c lp) so D{lp,nor-A-i(lp)) = d~\lm c D(lq,noT-A^(lp)). On the other hand, by Theorem 3.5 again, D{lv noT-A-*(lp)) cz d-*ln for any q e (0, oo]. W i t h d a s i n P r o p o s i t i o n 6.1, a n d 0 < g ^ l ^ ^ ^ o o , by either Corollary 3.7 or Corollary 4.5, with A = lp. As will be shown later, there are matrices A (the Cesaro matrix is one) for which D(lp,noT-A-*{lp)) is not a diagonal copy of lm for any p > 1. Thus the existence of a q < p such that D(lq, nor-A^ilp)) = d'1.^ does not imply Thus, one 'strong' converse of Proposition 6.1 is false, for p, A unrestricted. We do not know about the obvious, ordinary converse, which is the following: 6.2. Suppose either 0 < ^ < l or 1 < p < oo, and let d be the sequence of lp (quasi) norms of the columns of A. Does PROBLEM \0<q<p imply D{lpynoT-A-\lp)) = d~\l^. LOWER-TRIANGULAR MATRICES 109 REMARKS, (a) If p = 1, then the answer to Problem 6.2 is 'yes' by Corollary 4.9 (the implication is true because both hypothesis and conclusion are true). (b) If p = oo, then the answer to Problem 6.2 is 'no', as will be established later in Example 6.5. PROPOSITION 6.3. Suppose 1 < q ^ oo. Then D(lg, noi-A-*(ln)) = {b E co | {2£U (ank | bk \ )*}n e Q. Proof, b e D(lq, nor-A' 1 ^)) if and only if ADm maps lq into l^. When q = 1 the conclusion follows from a well-known characterization for matrix maps from lx to /«, [16, p. 167]. For other values of q the proof can be easily completed by using Holder's inequality and the Banach-Steinhaus theorem. COROLLARY 6.4. Suppose 1 < q < oo, and dk = sup^fca^. D(lq, nor-A-^U) = d-\lx if and only if {££=1 {ankdk-^}n Then e *«,. Proof. The proof is straightforward from Theorem 3.5 and Proposition 6.3. EXAMPLE 6.5. Let ank = (n — k + 1)'1, for n ^ k. Then dk for all k e N. For all q E (1, oo), k=l k=l k=l the nth. partial sum of a convergent series, since q' > 1. Thus for all q e (1, oo), and thus for all q E (0, oo). For q = oo, the wth partial sum of a divergent series. Thus i)(/00,nor-^l~1(/<a)) is not d^.Zoo, nor any diagonal copy of l^. The following lemma generalizes the usual characterization of compactness in lp, for 0 < p < oo, and c0. 6.6. Suppose A is a normal subspace of co, || • || is a normal quasinorm on X, and (A, || • ||) is AK and complete. Then F ^Xis compact if and only if F is closed and bounded, and LEMMA p| n-*<x> xeF 110 P. D. JOHNSON J B AND R. N. MOHAPATRA Proof. Let i f ^ 1 be such that ||* + y|| ^ M{\\x\\ + ||y||) for all x,y e A. Note that | | » - y | | > M^WxW -\\y\\ for each x,y e A. Since || • || is normal, sup^g F \\x — Pn{x) || is non-increasing with n. Suppose sup xe F || x — Pn{x) || ^ 4e > 0 for all n. Using the assumption that (A, || • ||) is AK, determine inductively a sequence {x{k)} <= F, and an increasing sequence {nk} of integers, such that ||a;(fc)— Pnk(x{k))\\ ^ 2e for all k, and \\xlk)-Pm(x{k>)\\ < eM'1 for all m > nk+1. Since y-Pn{y) e N(y) for all n e N and yew, and since || • || is normal, for m > k we have (by the normality of || • ||, again, since m ^ fc+ 1 implies nm > nk+v so a ^ ^ - P J ^ ) ) e N{x™-PntJx™))) > 2eM-1-eM~1. Thus no subsequence of {x{k)} is a Cauchy sequence, so JP is not compact. Clearly if F is compact, F is closed and bounded. Now suppose F is closed and bounded, and *ooxs F As mentioned in § 1, A being normal, with a normal, Hausdorff topology, implies coordinate projections are continuous on A. Thus a n = sup|a; n | < oo xeF for each n e N. Let S = {n e N: en G A); note that A a normal subspace of o> implies xn = 0 for all n $ S, x e A. For n e N\S, define | | e j | = 1, for simplicity in what follows. Suppose {x{n)} is a sequence of elements of F. Since F s n * - i U(an)> with U(an) the closed disc (or interval) of radius an about the origin in the field, and this product is compact in the topology of coordinatewise convergence, a metric topology on co, {x{n)} has a subsequence convergent coordinatewise to some yew. Changing the notation, let us use {x{n)} to denote this subsequence. Suppose e > 0, and let n be such that sup||a:-P n (a0|| < e. xeF LOWER-TRIANGULAR MATRICES 111 Then ||a;(«)-aj(*>|| ^ M(\\Pn(xim)-x{k))\\ r » L L <=i < M» Because the sequence {#(7l)} converges coordinatewise, the sum can be made as small as desired, for fixed n, by taking k, m large. Since e was arbitrary, this shows that {x{n)} is a Cauchy sequence in (A, ||*||). Since (A, || • ||) is complete {x{n)} converges to a limit (to y, in fact) in (A, || -||). This limit is in F, since F is closed. Thus F is compact. Suppose A is a subspace of a>, and T is a t.v.s. topology on A. Suppose Pn(A) s A for each n e N. Define AA = {x e A: Pn{x) -» x as n -> oo, in (A, T)}. Clearly AA is a subspace of A. If A and T are normal, AA is the closure of <pnX in (X,T) [11, Lemma 3.7], THEOREM 6.7. Suppose (A, ||-||A) and (fx, || -||A) are normal, normally quasinormed FK spaces. Suppose that (A, || • ||A) satisfies the conditions of Theorem 3.5, and <p <= p. Then, with respect to the operator quasinorm on D{ix,noT-A-*{\))A c {6 e w: ADb determines a compact map from p, into A}, with equality if (A, ||-||A) is AK. Proof. Suppose b e D(JX, nor-A-^X)^. By Proposition 2.5, it follows that ADb 6 ^(fi, A) is the limit, in the operator quasinorm of the sequence {ADPn{b)}n c= SC((JL, A). Each ADPn{b) has finite rank, so ADb is compact. Now suppose (A, || • ||A) is AK, and ADb e S£?{\JL, A) is compact. By Lemma 6.6, applied with F being the closure in (A, || • ||A) of ADb({x E /X : || x ||A < 1}), we have, sup || (/ - Pn)ADb(x) ||A ^ sup || (/ - Pn)y ||A -> 0 as n -> oo. veF \\\\l We claim that, for each n, sup ||(7-P n )^ J D 6 (a;)|| A = sup | | ( / - P n ) A D m \ x \ l l l l H | | 112 P. D. JOHNSON J R AND R. N. MOHAPATRA a consequence of the normality of A, /z, ||-||A, H'H^, and the non-negativity of the entries of A. For m> n, the rath entry of (I-Pn)ADb(x) is Vk-i<*>mkKxk> a n d o f ( J - p J ^ > i 6 i M is Z^i^m&IV&l, for x E fx. Thus because amk ^ 0, (I-Pn)ADb(x) e N((I-Pn)ADm\x\), for each x e [x, so the normality of || • ||A implies sup \\(I-Pn)ADb(x)\K sup ||(7-Pn)4D|W|a;|||A. lll^l lllll The reverse inequality is proved using the normality of \x and || •fl^,by the argument in the proof of Proposition 2.5 used to show that \\ADb\\* For x E fx,m ^n, the rath entry of AD\b_Pn{b)\ \ x | is zero; for ra > n, the rath entry is X&U+ia m k \b^c k \ ^ £2Ua mfc |& fc a fc |. Thus AD{b_Pnm\x\EN((I-Pn)ADw\x\) for all x E /x, so, b y the normality of || • ||A, sup M-D| 6 - Pn ( 6 )||cc||| A ^ sup || (I -Pn)ADm lllll | x | ||A ll||l = sup || (7 - Pn)ADb{x) ||A -* 0 as n ^ oo. The number on the left above is, by the proof of Proposition 2.5, the operator, quasinorm of ADb_Pnlb). Then, by Proposition 2.5, b — Pn{b) -> 0 in DifjbjiiOT-A^X)) as n -> oo, so 6 e D([x, nor-J.~1(A))A. COROLLAJRY -^lp)) 6.8. For 1 ^ p < q < oo, D{lq, nor-^" 1 ^)) is AK. is AK for every p E (0,OO). Proof. For q < oo, the conclusion follows from the preceding theorem and the fact that every continuous map from lq to lp is compact, for 1 ^ p < q < oo [19, Theorem A2, p. 206]. For q = oo, the result is less deep. Note that if i f is a matrix with non-negative entries, and lx c dom M, then, with U^ denoting the unit ball in l^, M{UV>) £ N(M(e)); the verification is trivial. Now, suppose b E D^^noT-A-1^)). Then 1 |6| e 7)(?00,nor-^4~ (^)), so ADm maps £w into lp. Since 0 < p < oo, the normal hull of any sequence in lp is compact in lp, so ADW{U9) <= N(ADw{e)) is relatively compact. By the preceding theorem, it follows that \b\-Pn\b\ = \b-Pn(b)\-+O in D ^ , noT-A'^lp)) as n -> oo, and thus 6 — 7^(6) -> 0 in DQ, since the topology on the latter is normal. LOWER-TRIANGULAR MATRICES 113 For p = 1, the conclusion of Corollary 6.8 follows from Corollary 4.9 and the fact that an FK space is an FK space under one topology only. REMARK. C O R O L L A R Y 6.9. Suppose that either l^p<q<coorO<p<q The following are equivalent: (a) D(lq, nor-^4~1(Zp)) = b.lmfor some b e a>; (b) only finitely many columns of A are in lp; (c) D{lq,noT-A~x{lp)) is finite-dimensional. = oo. Proof. Recall that S = {n:eneD(lq,noi-A-i(lp))} = {n: the nth. column of A is in lp} = N\{n:xn = 0 for all x e D ^ n o r - ^ - 1 ^ ) ) } . Let d denote the sequence of lp norms of the columns of A. If (b) holds, then S is finite, so D(lqynov-A-X{lp)) = sp{en: n e S) in view of the fact that N\S = {neN:xn = 0 for all x e Z>(Zg, n o r - ^ " 1 ^ ) )}• Thus we conclude that (c) holds. If (c) is true, then, since the en are linearly independent, we conclude that S is finite, whence (b) is true. Thus (b) and (c) are equivalent. If (b) holds, then d'1 e <p, so d'H^ = sp{en: n e S) c The reverse inclusion holds by Theorem 3.5. Now suppose (a) holds. Then, by Theorem 3.5, By Corollary 6.8, D(lq, nor-A'1^)) is AK, with the operator norm or quasinorm; D(lq, nor-A'1^)) is also an FK space, by Corollary 2.3. But d-i.ln is an FK space with a norm inherited from £«,, and is not AK if d~x has infinitely many non-zero entries. Since an FK space is an FK space under one topology only, it follows that d'1 e 9?, which implies (b). REMARKS, (i) Since (b) of Corollary 6.9 does not mention q, the three equivalent statements of that corollary are equivalent to the same three statements quantified by 'for all q > p\ in the case where 1 ^ p < 00. (ii) If lp,lq in Corollary 6.9 are replaced by arbitrary normal subspaces [A and A of o>, with <p c= p, then (b) and (c) are equivalent, and imply (a). (iii) Corollary 6.9 says that, for 1 ^ p < q < co, or 0 < p < q = 00, in no non-trivial case is there a best possible HPD inequality \\A\bx\\\p^K\\x\\q. 6.10. Suppose that either 0<p<q = ooorl^p^q^co, and the sequence d of lp norms of the columns of A is bounded away from zero. Then A does not map lq into lp. COROLLARY 5388.3.41 H 114 P. D. JOHNSON J B AND R. N. MOHAPATRA Proof. By the hypothesis, dr1 e £«,. Suppose q > p. Then If A maps Zg into lp, then (4£d-i)Zg = A(Dd-i(lq)) S 4 ( y g Zp, so, by Theorem 3.5, D ^ n o r - ^ " 1 ^ ) ) = eZ"1^. By Corollary 6.9 only finitely many columns of A are in lp; but then Aen $ lp for some (in fact infinitely many) n, contradicting the assumption that A maps lq into lp. Theorem 6.7 leaves unanswered the question as to whether D(/u,, nor-^4~1(Z00))A = {b e to: ADb determines a compact map from [x to lw} for JX a normal, normally quasinormed FK space containing <p. We do not know the answer to this question in general, but we do for \x = lq, with 0 < q ^ 1, and for a class of spaces [x including the lqi for 1 < q ^ oo. LEMMA 6.11. Suppose M = (mkn) is a matrix with non-negative entries, A g o>, fx is a normal subspace of to, /* £ dom(ilif), and M(/a) c A. iHffc(Ax)c i^35, awd /or every x e ft, y e \x, (Mx, y) = (x, Proof. Suppose x e n and y e Ax. Then | x \ e //., since /x is normal, and likewise |y| e A*. Thus, since M\x\ e A, oo > (M\x\,\y\> = S(S*»fc»l*J)ly*|. Since the m^^j are non-negative, it follows by reversing the summation order that oo > 2n(Sfc™fcj2/fcl )\xn\ > 2«IS*"**»#*IKI = <|a;|,|-ftfV|>, so Mly c fj,x} a n ( j that (Mx,y} = (z.-Mfy). If JU, is an FK space, then, as in the proof of Corollary 4.3, <•,?/> is a continuous linear functional on \x for each y e /xx. If <p £ /LC, the map V -*• ('>y) oi fxx into /a' is an injection. In this case, we will consider \& to be a subset of/x'. If (/*, || • H^) is a quasinormed FK space, denote by ||*||^ the dual norm on [i!. LEMMA 6.12. Suppose (/x, || - H^) is a normal, normally quasinormed FK space, and <p g fx. Then (ixx, || • ||^) is an FK space, and || • ||^ is a normal norm on JXX. Proof. By Proposition 2.2,/ux = D(fx, lt) is an FK space with the operator norm inherited from ££{\x, IJ, and the operator norm is normal on fxx. The normality of /x and ||«||A ensure that that operator norm is the same as the dual norm || • ||^ on fxx. LEMMA 6.13. Suppose M = {mkn) is a lower-triangular matrix, (//,, ||*||) is a normal, normally quasinormed FK space, <p g [x, and M maps fx into l^. LOWER-TRIANGULAR MATRICES Let Mn = || Men W^. If M is a compact map of p into l^, then {MJeJ-^ECQ. 115 Let us adopt the rule that s(n) will be written in place of sn whenever n is replaced by a more complicated expression. Proof of Lemma 6.13. If {Mn || en H"1^ $ c0, then there exist e > 0 and an increasing sequence (nr) s N such that M{nr) \\ e(nr) H"1 ^ 2e for all r. Replacing (nr) if necessary by a subsequence defined inductively, we may suppose that max \m(k,nr)\ r Now for each r E N, set Then, x and we claim that no subsequence of (x{r)) is a Cauchy sequence in ?«,. For r < s, Wfir) —X /y(8) II & "> rnn-v I f^ —Jih. <y(s) I \\X IIla/A. \JJU it ll ^ i la n» i / = max I x1^ I (since x^] = 0 for k < n8) nr^k<n, ^ m a x I x^ I )ll"1 max \m{k,nr)\ REMABK. The converse of Lemma 6.13 is false. Take p = lX} ||*|| = ll*lloo> and M to be the Cesaro matrix of order 1. LEMMA 6.14. Suppose (/x, ||*||) is a normal, normally quasinormed FK space, <p c. fx, M is a matrix with non-negative entries, and M maps /x into ?«,. Then the operator norm of M is the same as the operator norm of Proof. Note that Mt maps lx into fxx, by Lemma 6.11, and is a continuous map, with the dual norm ||*||^ on fjux, because (/xx, ||*||^) is an FK space, by Lemma 6.12. The operator norm of M is, by Lemma 6.11, sup ||J/a;IU = sup \(Mx,y}\= sup |< lll the operator norm of Mt. 116 P. D. JOHNSON J B AND R. N. MOHAPATRA 6.15. Suppose (/*,, || • ||A) is a normal, normally quasinormed FK space, <p £ /n, and (fxx, || • || ,) is ^4iL Then .D^nor-^- 1 ^))^- as £&e se£ THEOREM {b E a): ^42)5 determines a compact map from fx into £«,}. Proof. The inclusion c has already been proved, in Theorem 6.7. Suppose ADb determines a compact map from p into lx. Then so does ADW, by the following argument. Let yn be such that \yn\ = 1 and ynbn = \bn\ for each n e N. Since /z is normal, Dy maps /A into /A. Thus ADm = ADbDy is the composition of a continuous and a compact linear transformation and is thus compact. By Lemma 6.11, (AD^ = DmAk maps lt into \ix, and is the restriction to lx of the operator adjoint of ADW £ SC(fx, ?«,). Note that for continuous transformations from quasinormed into normed spaces, it can be shown that taking of adjoints preserves compactness. Since, by Lemma 6.12, fix is a closed subspace of /z', it follows that DmA* effects a compact map of lx into fix. By Lemma 6.6 applied to (fix, ||*||/) a n d * n e closure of the image of the unit ball in lx under the mapping DmAt, it follows that sup \\{I-Pn)DwA\x)\\, = sup \\Dlbl_PnmAt(x)\\fl, -> 0 as n -> 00. lxll^l HllSl That says that the operator norm of D|6|_pn|b|^4t goes to zero as n -» 00, and by Lemma 6.14, it follows that the operator norm of AD]b]_PnW goes to zero as n -» 00. By Proposition 2.5, \b\ — Pn\b\ = \b — Pn{b)\ -> 0, as whence, since the norm on D{fx, nor--4~1(Z00)) n -> 00, in D^noi-A-1^^)), is normal (Corollary 2.3), b — Pn(b) -» 0 in Z)(JLI, nor-^4~1(Z00)) as n -» 00, so 6.16. For 0 < q ^ co, D(lq, nor-A- 1 ^))*- is the set {b G co: -4i)6 determines a compact map from lq to l^}. COROLLABY Proof. For 1 < q < oo, the conclusion follows from Theorem 6.15, with /"• = ^» II'11^ — W'Wq- Suppose 0 < q < 1. The inclusion £ is known from Theorem 6.7. Let d be the sequence of £«, norms of the columns of A. By either Corollary 3.7 or Corollary 4.5, D(lg, nor-A' 1 ^)) = d'1!^, so, by topology comparison, D(lq, nor-^4~1(Z00))A = d~xCQ. Now suppose ADb determines a compact map from lq to l^. Then by Lemma 6.13, with p = lq, {\\ADb(en)\Qnec0, which says precisely that b e d~1c0. (Note that, since ADb maps lq into l^, dn = co implies bn = 0.) Thus b e D{lq, nor-A-1^))*. REMARK. AS may be clear from Lemmas 6.11 and 6.12, and the proofs of Lemma 6.14 and Theorem 6.15, under certain conditions on (A, ||-||A), LOWER-TRIANGULAR MATRICES 117 (fi,||'||^), an HPD inequality ||^4|6x|||A ^KWxW^ will imply a dual inequality H&.^lylll^a; < J£||y||Asc (same K, same &), and in the presence of perfection and reflexivity, each inequality implies the other. We will make no attempt here to uncover general conditions under which this duality is exploitable, but note for practical purposes that an inequality involving an upper-triangular matrix may be approachable through an inequality involving a lower-triangular matrix, and vice versa. In the final sections of this study, application of the results preceding will be made with A confined to certain classes of matrices. 7. Modified Nbrlund method For b e a), let Tb be the lower-triangular matrix each of whose columns consists of the sequence b, starting from the main diagonal. That is, the (m, n) entry of Tb is bm_n+1, for m ^ n. The main importance of these matrices is in this: suppose a> is identified with the space of formal power series, with yew corresponding to 9V — S £ U Vn7^'1 > denote by gb.gv the formal-power-series product of gb and gy, and let * denote the same product carried back to a>, t h a t is, & * V = {Sn=l K-n+lVrHim \ t h e n ° * V = Tb(V) f o r e a c h V e co- F o r 9 = 9b> W e denote Tb alternatively by Tg, risking ambiguity for convenience. Throughout this section, g will denote a formal power series, g(z) = S™-i ^nzn~xy n o * to be confused with b. If &! # 0, the condition for Tb to be invertible, then Tb~l = Tg-1 = Tg-U with g~x denoting the formal-power-series inverse of g = gb. If g represents some function analytic around 0, then g~x represents the reciprocal of that function. Let a denote the forward shift. From Theorem 3.5, we have the following: 7.1. Suppose (A, ||»||A) and (/x, ||-||A) satisfy the hypothesis of Theorem 3.5, and, in addition, A is invariant under a. Suppose b e A is a non-negative sequence, with bt > 0. Then COROLLARY D(n,nor-Tb-H\)) £ {||a^" 1 ^)IIA" 1 IK 1 1 ^ , with equality if and only if Tb maps {|| a^'^ib) l^"11| en H^./x into A. The proof is straightforward from Theorem 3.5 and the observation that the rath column of Tb is a**1-1^). COROLLARY 7.2. If, in addition to the hypothesis of Corollary 7.1, a is an isometry of (A, ||-||A) into itself, and ||e n ||^ = 1 for all n e N, then D(fi,nor-Tb-X(\)) with equality if and only if Tb(fi) c= A. g: I,,, 118 P. D. JOHNSON J B AND R. N. MOHAPATRA PROPOSITION 7.3. Suppose b is a non-negative sequence, b± > 0, and A is a normal subspace of m. If Tb{X) s= A, then A= Proof. Tb{\) c A implies A c ^ ( A ) , so A c nor-^-^A), since A is normal. Now, suppose x e nor-T6~1(A), that is, Tb\x\ e A. Clearly so | a; | e A, whence x e X. The conclusion of Proposition 7.3 does not imply that is l^. Take A = o>, for instance. However, £«, <= D(A,A), since A is normal. PROPOSITION 7.4. Suppose b e lx, and (A, ||-||) is a normed FK subspace of a) such that a is a continuous map of A into A with operator norm not greater than 1. Then Tb determines a continuous linear map of A into A. Proof. By the hypothesis, 2n=i ^n0"71"1 converges absolutely in j£?(A,A). For each x e w, {T^=i^n°:n~1) (x) converges coordinatewise to Tbx. Since the coordinate projections are continuous on (A, ||*||), it must be that the continuous linear map from A to A to which S"=i bnon~'y converges is Tb. The conclusion of Proposition 7.4 does not hold if ||«|| is required only to be a quasinorm. For instance, if b e l±\ Uo<p<i^v * n e n Tb does not map lp into lp for any p e (0,1). REMARK. COROLLARY 7.5. / / (A, || • ||) satisfies the hypothesis of Proposition 7.4, A is normal, and b e ^is non-negative with bx > 0, then nor-T&~1(A) = A. Proof. The proof is straightforward from Propositions 7.3 and 7.4. COROLLARY 7.6. Suppose g is analytic, g(z) # 0 for \z\ ^ 1, and b elv Suppose (A, || • ||) satisfies the hypothesis of Proposition 7.4. Then Tg determines an isomorphism of A into A. Proof. It is known (see [25, p. 246, lines 21-23]) that hypotheses of the corollary on g ensure that \/g has similar properties. So both Tg and Tg-1 map A continuously into A, by Proposition 7.4. PROBLEM 7.7. Is Proposition 7.4 true if the assumption that the coordinate projections are continuous on (A, || • ||) is omitted ? When A = Tb, where b e lx is non-negative with bx > 0, Corollary 5.9 may be forgotten in favour of Proposition 7.4 and its corollaries. However, in some cases Theorem 5.8 can still be useful in providing candidates for LOWER-TRIANGULAR MATRICES 119 the constants in the HPD inequalities arising from elements of D(lp,noi-Tb-%)). PROPOSITION 7.8. Suppose b elx is non-negative, and bx > 0. Then for 1 ^ p ^ oo, (33) ll^,|a:|||^(||6||1)(||a;y, is a best possible HPD inequality. The constant on the right-hand side of (33), viz. (|| bx ||), is the best possible constant. Proof. If 1 ^ p < oo we have, by Holder's inequality when p > 1, and trivially when p = 1, n I,bn+1_k\xk\ \p-l Hence M/v-lll \p — V , Jb\xW\p) — 2a n=l fc=l 00 00 n=k If p = oo, we have whence To show that the constant is the best possible, take xk = It is easy to verify that, with this definition, as N -> oo. On taking gr(z) = exp(z) and g(z) = (1 — rz)-1 in Proposition 7.8 we have the following two best possible HPD inequalities. 120 P . D. JOHNSON J B AND R. N. MOHAPATRA EXAMPLE 7.9. For 1 ^ p < oo and um ^ 0 (w = 1,2,...), £ ( £ u k / { m - k ) \ ) P < e* £ M^P, (i) oo (ii) Im \p I 1 \p oo Sf«* Sr^uJ < h—^ 1 r m«=l U=l / ZV- X — I m=l The constants on the right-hand side of each of the above two inequalities are the best possible. Note that condition (21), for (^4,/) to be an MDP triple, is precisely the condition that DfA satisfies (28) with K = Kv Thus if a non-negative sequence 6, with bx > 0, is non-increasing, Tb satisfies (28), and if A = DcTb for some positive CGW, then {A,c~x) is an MDP pair. By Corollary 5.7, we have, in this case, for 1 < p < oo, CP-^V))-1 e D(lp,noT-A-%)), since the sequence on the main diagonal of A, the a of Corollary 5.6, is 61c. This result is trivial if c ^ lp. These observations apply to the last example considered by Petersen [18], in which amn = (mim-n+l))-*, for n ^ m; that is, A = DbTbi with bn = wr*. Application of the conclusion in the paragraph above, after replacement of Jfc(6?J) by the equivalent sequence {w1~*p}m, yields {m~*} e Z)(^,nor-J.~1(^)), or, in inequality form, for some G, and for 2 < p < oo. Petersen obtains the same inequality for the larger range 1 < p < oo by noting that (-4,/,/" 1 ) is an MP triple w i t h / n = n, and applying what is here called Corollary 5.3. I t is salutary to note that a much better inequality (Proposition 7.11, below) can be obtained by a refinement, specific to this matrix, of the proof of Theorem 5.8. We shall need the following lemma. LEMMA 7.10. For n ^ 1, 1 < p < oo, LOWER-TRIANGULAR MATRICES 121 The lemma can be verified by using standard methods and the inequality (see [17, p. 191]) n+l TO=1 PROPOSITION 7.11. For un ^ 0 and 1 < p < oo, l/p is a positive constant which depends on p. Proof. Starting off by using Lemma 5.1, we have m \n=l n 11 = 1 TO U=i / n TO n-\p \p-l / n m \p-l 1)-*%I 71 = 1 1 y /,n-k+l)-*ui 71=1 \A:=1 / °° \7l=l = C(p) •r •^n 2 [w*p(wi-n + 1)*]-1 (£(%-*+:L)-i \&=i TO \p-l 2 (wi(m — n 4-1 • ))-*un\ um, TO=1Vn . < = l by Lemma 7.10. Now apply Holder's inequality and divide both sides of the inequality by ( 2(m(m-w + l)-*tt )») TO \l/p' n 71=1 / to achieve the result (assuming un > 0 for some n). The inequality is strict for u ^ 0. To illustrate the use of Theorem 3.5, we will show that the inequality of Proposition 7.11 is not the best possible and, indeed, that there is no best possible inequality of the form REMABK. \\A\bx\\\p^K\\x\\p, or 1141*111, for the matrix A of Proposition 7.11 and 1 < p < oo. 122 P. D. JOHNSON J R AND R. N. MOHAPATRA LEMMA 7.12. Set dPn= £[ro(TO-n + l)]-»»\ Then (34) d£ < Cn1-?, (35) dJ<C Kp<2, , 2 <p <oo, with C being a positive constant depending only on p. The lemma can be verified by familiar methods of estimation. PROPOSITION 7.13. Suppose amn = (m(m-n+l))-i for m^n, and mn = Oform< n. Then for 1 < p < oo, D(lp, noi-A-1^)) is not a diagonal copy of ln. a Proof. Suppose first that 1 < p < oo, and set ( oo \l/p 2 [ ( l ) ] * ) the lp norm of the nth column of A. By Theorem 3.5, it suffices to show that ADd-i does not map lp into lpi or, equivalently, By Lemma 7.12, d < 6"1, where (37) bn =TO<P-»/P= nlfP\ ioTl <p <2 (38) 6B = w*(log(w+l))-*, forp = 2, and (39) bn = n*, for 2 < 2? < oo. 1 Since d < 6" , b < d~l, so it suffices to show that 6 j p since the latter is a normal subspace of o>. Now, 6 e D{lp, nor-A-^lp)) if and only if ADb{lp) c Zp> if and only if i)6i4t(ZJJ.) c lp, (by Lemma 6.11). Thus, it suffices to find x e Zj,» such that Dfr4t(a;) ^ Z^. We take a; = {n-*}, where a = 2/p' in cases (37) and (38), and a = l/p' + | in case (39). With the x's as mentioned above one can verify that for 1 ^ p < oo, DbA\x) $ lp. PROBLEM proof. 7.14. Suppose A is as in Proposition 7.13 and d is as in the LOWER-TRIANGULAR MATRICES 123 (a) For 1 < p ^ oo, for which q e {l,p) is (b) Do there exist p e (l,oo), b e c0, and K such that 1 a: | H ^ £ | | 6a: ||p for all x e w ? (c) Is e maximal (with respect to <) among the non-decreasing positive sequences in D(lp,nor-A-X(lp)), for each ^) e (l,oo) ? REMARK. If the answer to (c) is 'no', then the answer to (b) is 'yes'. 8. Matrix A = DaJ The results of § 5 and the remarks of § 7 are especially applicable in the case where A = DaJ, with a e w (a > 0). (Note that the (C, 1) matrix is of this form, with an = rr1.) In this case {A,^1) is an MDP pair, and (A,a'1,a) is an MP triple; the conclusions deduced from Corollaries 5.4 and 5.7 are that aP-*{J*(aP))-* = {an*-1(S£-»«*p)-1}» e D{lptnor-A-*{lp)), for 1 < p < oo. The trivial result obtained when a $ lp is the best possible, for in this case nor-A-1^) is itself trivial. When a e lx and satisfies (*) am ^ Kant for n^m, which ensures that A = DaJ satisfies (28), then Corollary 5.9 says that J^a)'1 G D(lp, nor-A-itfp)), with 1 < p < oo, for </fc(a) is the sequence of lx norms of the columns of A. This result is never better than that in the preceding paragraph; that is (^(a))" 1 < aP~1{Jt{av))~1, meaning al-vJ^aP) {J^a))-1 is bounded, if a e lt and satisfies (*), for p e (l,oo). To see this, by (*) we have, for m ^ n, Hence which yields the result. The inverse of A = DaJ is A'1 = J^D^i, the nth. row of which has two non-zero entries, —cbn^1 and a^1, for n > 1. From Corollary 3.8 and the remarks above, we have the following: PROPOSITION 8.1. Suppose 1 < p < oo, a is a positive sequence, a e lp, and A = DaJ. If then 124 P . D. JOHNSON J B AND R. N. MOHAPATRA Proof. We have The nth entry of oP-^J^aP))-1 is Clearly ££=* K / a n ) p ^ 1, so, since ( S = n ( f l A f t follows that a1P-\Jt{av))-1 is equivalent to a,'1. Thus is bounded, it We get the result from the first clause of Theorem 3.5 since the d of that theorem is equivalent to a. COROLLARY 8.2. / / 1 < p < oo, a is a positive sequence, a e lp, and Proof. Setting A = DaJ, we have cr 1 e D^.nor-^l" 1 ^)), by the proposition, and thus ADa-i maps lp into Z^. Since a elp, ADa-i{a) = {nan} e lp. REMARK. The corollary obtainable from Proposition 8.1 and Theorem 3.5, that if {Em^n (am/an)p}n *s bounded, then a is equivalent to d = (J^aP)^, the sequence of lp norms of the columns of DaJ, is easy to prove directly. EXAMPLE 8.3. Set an = ((ft-1) \)~1 in Proposition 8.1. Since and 1 < p, we have The result derived from Proposition 8.1, a best possible HPt) inequality, is now given by m=l (( m t \V a \ r i^ ^ 0, 1 < p< oo, / or ( m n=l \p / REMARKS, (i) The constant O(^)) cannot be taken from either Theorem 5.2 or Theorem 5.5 because the sequence ax~vJt{aP) has been replaced by the equivalent sequence a in these inequalities, with no care taken about the constants involved in the equivalence. If we take the requisite care, LOWER-TRIANGULAR MATRICES 125 Theorem 5.5 gives, with Kx = I, (ii) If r e (0,1), and an = rn~x, then a satisfies the hypothesis of Proposition 8.1. However, the inequality derived from Proposition 8.1 is the same as that in (ii) of Example 7.9. The reason is that TaDa = DaJ, for = a^nor-T^y. this a, so nor-iDJ)-1^) (iii) In the case where {Sm=n (am/an)p}n *s n o * bounded, the observations in the proof of Proposition 8.1 can still be used to catch D(lp, noi-A'1^)) between two diagonal copies of lx, namely ap~1(Jt(ap))~1i00 and &?«,, with bn = (a^i + fl~p')1/p', for n > 1. For instance, if an = rrr, rp > 1, 1 < p < oo, we have, replacing ap~1(Jt(ap))~1 and b by equivalent sequences, (40) { m - % c D(lp, nov-(DaJ)-Hp) £ {m%. (iv) It is worth pointing out that we can get a better result than that given in (iii) by applying the first clause of Theorem 3.5, since by doing this, we can replace bl^ by dr^.l^ (with d as in Theorem 3.5). That this is a better result follows from \fc=n so that l/dn < \/an < b. r On taking an = n~ , rp > 1, 1 < p < oo, and following the steps as in (iii), we have (41) { m r - % <= £ ( ^ , nor-fZV)" 1 ^) c { m ' " 1 ' ^ . 9. cesj) When A is the Cesaro matrix, nor-^4~1(Z7,) goes by the name of cesp, for 1 < p ^ oo (see [21]). To simplify the discussion and to make contact with this notation, current in the literature, we will denote by cesj,r) the space nor-A-X{lp), with A = DaJ, an = n~r. Note that for 0 < p < oo, ces^r) is non-trivial if and only if rp > 1 (and, in particular, ces^ is non-trivial for all r ^ O ) . From the remarks of the last section, (42) {W"-% £ «*$"> for rp > 1, 1 < p < oo. For p = oo the inclusion (42) does not hold in every non-trivial case, since we require that r > 0. The inclusion (42) when rp > 1 continues to be true when p = 1. The following proposition gives the basic inclusion relations among the ces],r) and the lp. 126 P . D. JOHNSON J B AND R. N. MOHAPATRA 9.1. Suppose 0 < r < t, and 0 < q < p ^ oo. cesp g cesy, and the inclusion is proper if tp > 1. ces£r) c: ces*,r), and the inclusion is proper if rp > 1. If r < 1 and rp > 1, Jfan Zp is wo£ contained in cesj,r). J / r p > 1, $e?i ces*,r) is wo£ contained in l^ (and thus certainly not in lp). / / r ^ 1, 1 < p < oo, emd rp > 1, then lp c ces*,r). PROPOSITION (a) (b) (c) (d) (e) r) Proof. We first point out t h a t if 0 < p < oo, rp > 1 then {wa} e ces^r) if and only if a < r — 1 — p " 1 ; and if p = oo, r > 0 then {?ia} e ces^r) if and only ifa^r-1. (a) First suppose p < oo, since r < t, ( TO \ p n=l I TO / \n=l TO r) 0 which proves that ces^ c ces^ . Suppose tp > 1. If rp < 1, then cefl« = {0}, strictly contained in ces^ , so suppose r > 1/p. Integral comparison shows that {n'-1-™^} e ces^Xces^. Now suppose that p = oo. The inclusion ces^ £ ces^ is easy, and it is not hard to see that {n^1} e ces^Xces^' for any a e (r,t). (b) The inclusion is easy, by the definition of the ces£°, since lq^lp. Suppose rp > 1, and we may as well suppose rq > 1. If p < oo, then {n-«} e cesj,r) \ cesjr) for any a e (1 - (r - p " 1 ) , 1 - (r -q~ 1 )). If p = oo, then the same assertion holds if we replace 1/p by zero. (Details are left to the reader.) (c) Note that r < 1, rp > 1 imply 1 < p < oo. For p < oo, {n-^elpXcesg* -1 1 for any a e (p ,1 — (r— p" )]. In the case where p = oo,e e l^ces^K (d) Take a sequence tending to oo and separate the non-zero terms with enough zeros to make a new sequence x e ces^^Z^. (e) For l < p < o o , r ^ l implies lp c (nr-1}ZJJ c cesj,r), by the remarks preceding the proposition. The same holds for p = 1, r > 1, by Corollary 4.9 with fi = Zx; note that d"1 is equivalent to {W"1}. Conclusions (c) and (e) of the preceding proposition are cruder than our purposes require; the main query of this paper, applied to lp and cesj,r), is not just whether or not I lies in ces^r), but what diagonal copies of lp lie in cesj,r). However, we see that if 0 < p < 1 and rp > 1, then lp c cesj,r). For the proof of the above assertion we note that from Theorem 19 of [10], 0 In \p IS \Jfc-l 1**1/ fc-1 LOWER-TRIANGULAR MATRICES 127 and hence co I n=l I n \p oo n fc=l / n~l &=1 «ZI**l p Zn-*»£4 Sl**lp, fc^l n=& ft-1 for some positive constant A = A(p). THEOREM 9.2. Suppose 1 <p < oo, and rp > 1. TAew JD(?g, cesj,r)) is Twtf a diagonal copy of /«,, for I < q ^ p. Proof. In view of Theorem 3.5 it is enough to show that B does not map lg into lp where B = AD{nr-nP}. On considering Bin-01}, where gr 1 < a < 1, we get the desired conclusion. PROBLEM 9.3. Suppose 0 < p < 1, and rp > 1. For which g > 0 is D(Jg, cesj,r)) a diagonal copy of ?«, ? It is a consequence of Theorem 9.2 that Hardy's inequality is not a best possible HPD inequality, but the proof of this assertion rests so much on generalities that the careful reader might well ask to be shown an HPD inequality better than Hardy's. Here is one: Define the sequence b by the following: b2n = n (9i = l , 2 , . . . ) , bm = 1 form£{2«: n = 1,2,...}. Then with A, the (C, 1) matrix, and 1 < p < oo, for some C(p), for all x e lp (or, for all x e o>). We omit the details of proof. REMARK. The example in the paragraph above is somewhat unsatisfying, for although the sequence 6 with which we replace the e implicit in Hardy's inequality is manifestly larger than e, in the sense of < , it is not increasing, nor does it tend properly to oo. This leads to the following: 9.4. Suppose 1 < p < oo, and rp > 1. (a) Is {W"1} maximal (or equivalently, maximum) among the monotone positive sequences in D{lp, cesj,r>) ? (b) Does there exist b e D(lp, cespr)) such that {^n"^"1} e c0? PROBLEMS remarked after Problem 7.14, if the answer to (a) above is 'no', then the answer to (b) is 'yes', for r > 1. In the case where p = oo, the answer to (b) above is 'no'. If there existed such a 6, then there exists a positive such b, because D^, ces^) is REMARK. AS 128 P . D. JOHNSON J B AND R. N. MOHAPATRA normal. But if b is positive and {bnn1~r} tends to oo, and A = D{n-T)J, it is easy to see that the row sums of ADb tend to oo, so ADbl^ £ lx, so b PROPOSITION 9.5. Suppose 1 < p,q ^ oo, and rp > 1. Then a>r—l/p inclusion on the right is proper, the inclusion on the left is proper if Proof. Suppose b e D(lq, ces^,r)), and <x> r — \/p. We want to show that b e {na}lg>. Since both spaces in question are normal, we may as well suppose that b is non-negative. Qlq>, by 6eD(Zg,cesJ,r)) implies D{n-r)JDb{lq) clpi so D^D^-^lp.) Lemma 6.11. Since oc-r+l > 1 -l/p = l/pf, {w~a+r~1} e lp,, and therea>r-l/p, fore DbJtD{n-r){n-«+r-1} = DbJt{n-^+1^} e l^. Note that r > l/p imply a + 1 > 1. By integral comparison, therefore, J^wr""1} is equivalent to {n-*}. Thus Db{n~«} = b.{n-«} E If, so 6 e {nP^lg.. The inclusion D(lq, ces^r)) c: na>r-i/3j{wa}^' * n u s established is proper because the space on the right, being a descending intersection of FK spaces, has a natural FK topology (see [6, pp. 226-7]) which is not a norm topology (because it is not locally bounded; the verification is routine), while the space on the left is, by Corollary 2.3, a normal FK space. Since an FK space is an FK space under one topology only, the two spaces must be distinct, so the inclusion is proper. Suppose x e Iq., and p < oo. We want to show {nr~1/p}x e D(lq, ces^r)). As before, we may as well assume x is non-negative. By Lemma 6.11, it suffices to show that D{n'-up).xJtD{n-r} maps lp. into lq.. Suppose y e lp., = 1 . Then n=l ( ^n) '(( £ °° \ Q'/P I °° V mw—'Pi ZJ I m=n / ^ 2(^ r ~ 1/p C2 3 rn-rp)9lP \ Q'IP' ( V I ii \V'\ ZJ I Um I I \m=n I I L O W E R - T R I A N G U L A B MATRICES 129 r Now suppose p = oo. We want to show that [n }.z e D(lq, ces^). It suffices to show that D^^D^-,) maps Zx into lq.. A reprise of the preceding proof, replacing the use of the Holder's inequality by shows what has to be shown. In the case where p = q, the inclusion is proper because {if-*} e D(lp, ces¥)\{n'-vp}lp,. REMARK. Let us compare the inclusion (41) with the case q =p of Proposition 9.5. We remark that neither of (43) {n-%, K-"*%, is included in the other. Also neither of (44) {^-1/?%, 0 {n%- ot>r-l/p is included in the other. To justify the assertions made, we may note that if xn = nr~x then x belongs to the first of the sets (43) but not the second; if xn = [0 (otherwise), where r—1 < /? < r — l/p, then x belongs to the second set (43) but not the first. If xn = nr~llv then x belongs to the first of the sets (44) but not the second; if vn 0 ^ [71 — & , v — l , z , . . . ; , (otherwise), then x belongs to the second of the sets (44) but not the first. COROLLARY 9.6. 7/1 < p,q ^ oo, The inclusion on the right is proper, and that on the left is proper if p = q. Proof. Just take r = 1 in the preceding proposition. In the cases where p ^ q, we do not know about the strictness of the inclusion 5388.3.41 130 P . D. JOHNSON J B AND R. N. MOHAPATRA 9.7. Suppose 1 < p < oo and rp > 1. (a) For which q, with 1 < q ^ oo, is the left-hand inclusion in Proposition 9.5 strict? (b) For which q, with 1 < q < p, is D(lq, ces<,r)) AK ? PBOBLEMS An answer to (b) would have bearing on (a), because for 1 < q ^ oo, {nr~1/p}lg>, being a diagonal copy of l^, has a natural FK topology inherited from l^ with which it is AK, since 1 < q' < oo. REMARK. PROPOSITION not AK. in D(lp, 9.8. Suppose 1 < p < oo, and rp > 1. Then . D ^ c e s ^ ) is In particular, In*'1} is not the limit of its finite sections, (i^{?ir~1})(, ces^). Proof. Set x = {n*'1}. First suppose p < oo. Fix t e N and let Dt = Dx_Pi(x). Let A = D{n-r}J. By Proposition 2.5, it suffices to show that || ADt ||, the operator norm of -42^ as a map from lp to lp, is bounded away from zero as t ranges over N. Let y = 2&t^+i ek- Then \W\\y\\P t+m In -1 ( ( oo \p + « / t+m 2 ^"^ 2 *f"1) 2 / t+m ?&-»•*> \p\ 2 jfc*-1 1/JJ \p\ \n-4+m+l (G is a positive constant that can be chosen independently of m and t) m) Letting m -> oo, we have ||^AII ^ ^'» " ^ ^ ^ ' > ^> independent of f. Now suppose p = oo. Proceeding as before, but with y replaced by e, we have m supra-*" 2 m>t m>t sup(( with K independent of t. r-1 LOWER-TRIANGULAR MATRICES 131 The case where p = oo has been left out of Problems 9.7 because the answers for that case are known. PROPOSITION 9.9. Suppose r > 0; then D(lq, ces^) is not AK, for all 0 < q ^ oo. Proof. The case where q = oo is covered in Proposition 9.8, and the conclusion for 0 < g ^ 1 is a consequence of either Corollary 3.7 or Corollary 4.5, by which D(lQ, ces(r)) = {n*}^ for q e (0,1]; no such diagonal copy of lx can be AK with an FK topology. Suppose 1 < q < oo. By Proposition 6.3, {nr~1/Q'} e D(lq, ces^), for, in this case, S L i (°W I bk I Y = n~rq> 22-i ^ r " 1 < Cnr^n* = C, with 0 independent of n. Set a; = {n*-11*}, Dt = Dx_Ptix), and y = Proceeding as in the proof of Proposition 9.8, with A = D(n-r)J and denoting the operator norm on S?(lq, ?«,), we have > llyllf"MAylL = m-^Bup*-' S t+m sup w~r 2 m)-r({t + m Letting m -» oo, we have H-4DJ ^ C > 0, so, by Proposition 2.5, P^z) does not converge to a; in D(lq, ces^'). COROLLABY 9.10. For p = oo, 1 < q ^ oo, ^ e left-hand inclusion in Proposition 9.5 is proper. Proof: {nr-1+1/P']lq. = {nr}lg, is AK, whereas D ^ c e s ^ ) is not, by the preceding proposition. REMARKS. Referring to Theorem 9.2 and Propositions 9.8 and 9.9, we note that the spaces D(lp, ces^r)), for 1 < p < oo, rp > 1, and D{lq, ces^), for 1 < q ^ oo, r > 0, are normally normed FK spaces which are not AK, and also not diagonal copies of l^. Theorem 9.2 and Propositions 9.8 and 9.9 have the additional easy but interesting consequence that cesj,r) is not a diagonal copy of an lq, for 1 < p < oo, rp > 1. For r > l,ces[r) is a diagonal copy of lv namely 132 P . D. JOHNSON J B AND R. N. MOHAPATRA r 1 {??. ~ }Z1) by Proposition 4.8, and we do not know about ces],r), for 0 < p < 1, rp > 1. 10. ces[p,s] Suppose s = {sn} is a sequence of positive numbers, and Sn, = 2&-i sk for each neN. Define (N,s) = Ds-iJDs; that is, {N,s) is the lowertriangular matrix whose (m, n) entry is sn/Sm for n ^ m. If sn is a constant for each n then (N, s) reduces to the (C, 1) matrix. In conformity with the notation of [11], we define ces(j?,s] = nor-(N, s)~1(^), forO < p ^ oo. Thus ces[p,e] = cesp, e being the sequence each of whose entries is 1. Note that ces[#,s] is non-trivial if and only if S'1 e lp (see [11]). Since {N, s)"1 = .Ds-i(.Ds-u/)-1, it is easy to see that ces[p,s] = ^ whence D(fx, ces[p,s]) = s^Ddx, nor-(Z)/S-iJr)~1(Z1J)) for any /x£a>; thus information about D(fjL,ces[p,s]), for any positive sequence s, is interchangeable with information about D(/x, nor-^L" 1 ^)), for A = DaJ, with 'a' any decreasing positive sequence. Thus, for instance, using the fact t h a t 8-1 is decreasing, we have the following: PROPOSITION 10.1. For 1 < p < O O ^ - W - P ^ O S - P ) ) - 1 e D(lp,ces[p,s]). The proof of the proposition is clear from Corollaries 5.4 and 5.9, the remarks of § 8, and the remarks above. COROLLAHY 10.2. / / l<p<oo, s-leD{lp,ces[p,s]). and { S / ^ S t n W ^ l then Proof. The hypothesis says t h a t s" 1 < .s- 1 # 1 - p (J t (#- p ))-~ 1 , so the conclusion follows from the proposition and the fact t h a t D(lp, ces[^>, s]) is a normal subspace of m. Note t h a t Proposition 8.1, with a = S'1, gives the following: PROPOSITION 10.3. / / 1 < p < oo and {Snp 2 ^ = n £ m - p } e lM then Copson [2] proved the following inequality, which, by its form and generality, seems superior to Corollary 10.2 as a generalization of Hardy's inequality. LOWER-TRIANGULAR MATRICES THEOREM D [2]. Suppose In U=l n COROLLARY 133 1 < p < oo, 's' is a positive sequence, and \p\ l/p / / 10.4. For 1 < p < oo, s~1/plp ^ Proof. The proof amounts to writing down the norms naturally supplied to s~llvlp and nor-(JV, s ) - 1 ^ " 1 ^ ) by H'l^, and applying Theorem D (to which, except for the constant (p/{p — 1)), this corollary is equivalent). COROLLARY 10.5. / / 1 < p < oo and A = DsuPS-iJDs, S-VP e then D(lp,noT-A-Hlp)). This is another restatement of Theorem D. Note that in this statement, the matrix A varies with p, whereas in Corollary 10.4 the matrix involved, namely (N, s), is undisturbed by changes in p. REMARKS. Copson, in [3], proved two other inequalities that can be reformulated in ways parallel to the reformulation of Theorem D by Corollaries 10.4 and 10.5. In [2], Copson also proved the following inequality, for 1 < p < oo, un ^ 0: ( n / oo \p\l/p \k=n II Hardy [9] noted that this inequality does not really require a separate proof, but can be derived from Copson's first inequality (Theorem D) by what Hardy called 'the converse of Holder's inequality', the fact that lpx = lp. and \\x\\p. = sup nyu ^ | (x, y} |. This derivation is an instance of the phenomenon remarked upon at the end of § 6 here, whereby inequalities involving upper-triangular matrices are derived from HPD inequalities involving lower-triangular matrices by the taking of linear adjoints or transposes. Copson's second inequality in [2] can be seen (in several different ways) to be a transposed form of his first inequality. For fixed p e (l,oo), the constant {p/(p — l)) in Theorem D is the best that will do for all s and u, since for s = e the inequality becomes Hardy's inequality, for which it is known that (p/{p — 1)) is the best possible constant. It would be rather surprising if (p/(p — 1)) were the best possible constant for each p and s (letting only u vary) but we do not know an example of an s for which a better constant can be found. The following proposition, corollary, and example bear on the strictness of the inclusion in Corollaries 10.4 and 10.5. 134 P . D. JOHNSON J B AND R. N. MOHAPATRA PROPOSITION 10.6. / / some subsequence of s is bounded, and 1 < p < oo, then Proof. The problem is to find a non-negative sequence x such that 8"*x = {sn1/pxn} $ lm while LupSn-i I A Replacing x by s~xx, it suffices to find a non-negative x such that sW(N,s)(x) = Sfc-i^jJn e Zp. Note that s^S'1 e lp; this is a consequence while {s^S^1 of Corollary 10.4 (for, since en e nor-(^,s)-1(s~1/pZ1)) for each n, the nth column of (N, s) must lie in s~1/plp} for each n), and is provable directly as follows: because of the relationship of s and 8, for n > 1, we have oo r>ot 2 SJJSJT 1 < %~p dx < oo, k=n JSn-i where a = || 5 \\± (= oo if s $ ly). Now, if some subsequence of s is bounded, then some subsequence of 1/p ^ e > 0 for all k. Since s-vp' i s bounded away from zero, say sn- ' can find an unbounded sequence x with support (indices of sVpg-i elp,we non-zero entries) contained in {nk: k e N}, so that {sn~1/p'a;n} £ ?«> but with support so spaced out that fc-1 The details are omitted. EXAMPLE 10.7. Set sn = n\. In this case, for n > 1, 2+ £ Following the reduction in the proof of Proposition 10.6 and the conclusion of Corollary 10.4, we can show that 8r,«)-1(«-1'*IJ,) (1 < p< oo). A more general result throwing light on Example 10.7 is COROLLARY 1 < p < oo, 10.8. Suppose sequences s and S are equivalent. Then, for _ LOWER-TRIANGULAR MATRICES 135 Proof. Following Proposition 10.6 and Corollary 10.4, it is sufficient to show that, for x, a non-negative sequence, (8nv*8n-lJtxi\elp \ implies (sn-"*>'xn) e lp. k<=l I Since s and S are equivalent, s 1 ^ " 1 is equivalent to s~1/p', and surely for x non-negative x < J(x) = {££=1 xk}n, so sn~1/v'xn < s1/piS~1J(a;) and the conclusion follows. REMARK. 5 and 8 are equivalent if and only if there is a constant A > 1 such that 8n+1 ^ X8n (n e N). The following results, all corollaries of Theorem D, return to the question of when s" 1 ^ c= cea[p,8]. COROLLARY 10.9. If sn^ ( n Proof. We have n 1 for all n, then for 1 < p < 00, un ^ 0, / n \#\l/p U=l / / p ^ P—I n ip\i/p (Theorem D) using twice the assumption that sn ^ 1, and once the assumption that 1 < p < oo. p/{p — 1) is certainly not the best possible constant here for all s and p e (1, oo), since if sn ^ 1 + e > 1 for all n, then (1 + e)~1p/(p — 1) will also work. We do not know if p/{p — 1) is the best possible constant for each p e (l,oo) and s satisfying infnsn = 1. On the subject of best possible HPD inequalities involving the matrices (N, s) and the spaces lp, we submit the following remarks, in addition to Proposition 10.3. (a) For p < oo, the sequence of lp norms of the columns of (N,s) is d = siJ^S-P))1'*, so, for 0 < q ^ p < oo, by Theorem 3.5, D(lq, ces[p, s]) is a diagonal copy of ?«, (and d'H^ is the only possibility) if and only if REMARK. 136 P . D. JOHNSON J R AND B . N. MOHAPATRA (N,s)Dd-i = Ds-iJDUi(s-p))-iiP maps lq into lp. Sometimes this happens (Proposition 10.3) and sometimes not (Theorem 9.2, taking the r there to be 1, the s here equals e). (b) By Theorem 3.5 and Proposition 6.3, for 1 < q ^ oo, D(lg, ces[oo, s]) is a diagonal copy of l^ if and only if D(lq, ces[oo,s]) = s^Sl^, if and only (c) It is never the case that s"1 is a maximum sequence in D{lg, ces[p, s]), for 1 < p < co; for if D(lq,ces[p,s]) = s'H^, then, by (a) above and Theorem 3.5, s~1l00 = D{lq>ces[p,s]) = s-^J^S-^))-1^, so ces[>,s] is non-trivial, implying S'1 e lp, and s'1 and s-1(Jrfc(#-p))~1/p are equivalent. This cannot be, for S'1 e lp implies Jt{S-v)llv e c0. The same holds for p = oo, 1 < q ^ oo. If D(lq, ces[oo, s]) = s" 1 ^, then, by (b), above, s*1 and s~xS are equivalent. Thus S is bounded, so 5 e lv But then 8n~* S L i V ^ IIs I l i ^ / w -> oo as n-+co, so D(^,ces[oo,5]) is not a diagonal copy of ?«, after all, by (b). (Note that if 0 < q ^ 1, D(lq, ces[oo, s]) = s" 1 ^ if and only if s e llt by Corollary 3.7 or Corollary 4.5 and the fact that s"1 and s^S are equivalent if and only if s 6 lv) Added in Proof. In the proof of Proposition 1.1 one should use Cauchy nets in place of Cauchy sequences; but the proof remains the same. Theorem 9.2 can be extended with the same proof to 0 < q < p < oo. Hence Problem 9.3 is partially answered. Acknowledgement The authors are extremely grateful to the referee for his valuable comments, supply of some proofs, and constructive criticisms which led to the improvement of the paper. REFERENCES 1. T. A. BROADBENT, 'A proof of Hardy's convergence theorem', J. London Math. Soc. 3 (1928) 242-43. 2. E. T. COPSON, 'Note on series of positive terms', ibid. 2 (1927) 9-12. 3. 'Note on series of positive terms', ibid. 3 (1928) 49-51. 4. G. S. DAVEES and G. M. PETERSEN, 'On an inequality of Hardy's (II)', Quart. J. Math. Oxford (2) 15 (1964) 35-40. 5. E. B. 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WILANSKY, Functional analysis (Blaisdell, New York, 1964). 25. A. ZYGMUND, Trigonometric series, Vol. 1 (Cambridge University Press, 1959). Mathematics Department American University of Beirut Beirut Lebanon