MEASURE OF WEAK NONCOMPACTNESS AND REAL INTERPOLATION OF OPERATORS ANDRZEJ KRYCZKA, STANISLAW PRUS, AND MARIUSZ SZCZEPANIK Abstract. A new measure of weak noncompactness is introduced. A logarithmic convexity-type result on the behaviour of this measure applied to bounded linear operators under real interpolation is proved. In particular, it gives a new proof of the theorem showing that if at least one of operators T : Ai → Bi , i = 0, 1 is weakly compact, then so is T : Aθ,p → Bθ,p for all 0 < θ < 1 and 1 < p < ∞. 1. Introduction In 1960, M.A. Krasnoselskii [20] proved that under the hypotheses of the Riesz– Thorin interpolation theorem (i.e. if T is a linear operator such that T : Lpi → Lqi is bounded for i = 0, 1 where pi , qi ∈ [1, ∞]) and the additional assumption that T : Lp0 → Lq0 is compact and q0 < ∞, it follows that T : Lp → Lq is also compact. Here 1/p = (1 − θ)/p0 + θ/p1 , 1/q = (1 − θ)/q0 + θ/q1 and 0 < θ < 1. This has initiated a series of theorems which refer to the question whether in the above theorem the couples (Lp0 , Lp1 ), (Lq0 , Lq1 ) and the spaces Lp , Lq can be replaced by couples of Banach spaces (A0 , A1 ), (B0 , B1 ) and real interpolation spaces Aθ,p , Bθ,p . More generally, if T , viewed as a map from Aθ,p to Bθ,p , inherits any compactness properties which it may possess as an element of L(Ai , Bi ). Since the works of J.-L. Lions and J. Peetre [21], several authors have obtained results of different degrees of generality. Finally, M. Cwikel [15] showed that if T : A0 → B0 is compact and T : A1 → B1 is bounded, then T : Aθ,p → Bθ,p is also compact for all 0 < θ < 1 and 1 < p < ∞. The similar problem for weak compactness properties of T was studied as well. Here a key result is due to B. Beauzamy [8], who proved that if the embedding I : A0 ∩ A1 → A0 + A1 is weakly compact, then so is I : Aθ,p → Aθ,p . Next, S. Heinrich [18] extended this result to closed operator ideals. Other generalizations of Beauzamy’s result are due to M.Kh. Aizenstein and Yu.A. Brudnyi (see the book [9]), and to L. Maligranda and A. Quevedo [22] (see also M. Mastylo [23]) who established the following: if 0 < θ < 1, 1 < p < ∞ and T : A0 ∩ A1 → B0 + B1 is weakly compact, then so is T : Aθ,p → Bθ,p . In particular, this is the case if T : A0 → B0 or T : A1 → B1 is weakly compact. The Riesz–Thorin theorem gives also a logarithmically convex estimate for the norms of an interpolated operator. This motivated some authors to find quantitative versions of the above-mentioned results on compact and weakly compact operators. The main tools in such investigations are measures of noncompactness 1991 Mathematics Subject Classification. Primary 46B70; Secondary 46A50, 47B07. Key words and phrases. The Riesz–Thorin theorem, interpolation space, measure of weak noncompactness, w∗-cluster point of a sequence, free ultrafilter. 1 2 A. KRYCZKA, S. PRUS, AND M. SZCZEPANIK and weak noncompactness. Measures of noncompactness appear in various contexts, so it was convenient to define many different measures (see [1]). Here let us only mention the Hausdorff and separation measures. In contrast, very few measures of weak noncompactness can be found in the literature. Most authors use the measure of weak noncompactness ω introduced by F.S. De Blasi [16], which can be seen as a counterpart of the Hausdorff measure of noncompactness. De Blasi’s measure was successfully applied to operator theory and to the theory of differential and integral equations (see [5] and the references given there). The measure ω was also used by A.G. Aksoy and L. Maligranda [3] in order to obtain a quantitative version of the theorem on real interpolation of weakly compact operators (similar results for the Hausdorff measure of noncompactness were earlier proved in [27]). Recently the thorough study of this kind of estimates for operators under real interpolation was undertaken by F. Cobos, P. Fernández-Martı́nez, A. Manzano and A. Martı́nez in the series of papers [10]–[14]. In particular, some essential restrictions on interpolation of De Blasi’s measure are indicated (see [12], [14]). In this paper we introduce a new measure of weak noncompactness γ, which can be treated as a counterpart of the separation measure of noncompactness. Its definition bases on an idea due to R.C. James [19]. In the general case, the measures ω and γ are not equivalent. The measure γ appeals directly to the norm topology, while in the definition of ω the weak topology is involved. This seems to determine more geometric character of γ. However, γ can be expressed by various formulae. In particular, we show that γ coincides with the function based on the double-limit criterion of weak compactness, which was considered by K. Astala and H. -O. Tylli in [4]. Our main result shows that Riesz–Thorin-type estimates hold for the measure γ without any additional restrictions. Throughout this paper, by a Banach space we mean a real one. However, the results presented here can be easily restated and proved for complex Banach spaces. We denote the open unit ball of a Banach space X by BX and its closure by BX . Furthermore, conv A stands for the convex hull of a set A ⊂ X and the closure of conv A will be denoted by Conv A. For simplicity of notation we use the same letter to designate an element of a Banach space X and its canonical image in the second dual X ∗∗ . The abbreviations with the beginning w∗ refer to the weak-star topology. 2. Measures of weak noncompactness 2.1. Measure γ. Let (xn ) be a sequence in a Banach space X. We shall say that (yn ) is a sequence of successive convex combinations, or scc, for (xn ) if there exists pn+1 a sequence of integers 0 = p1 < p2 < p3 < . . . such that yn ∈ conv(xi )i=p n +1 for every n. Similarly, vectors u1 , u2 are said to be a couple of scc for (xn ) if u1 ∈ conv(xi )pi=1 , u2 ∈ conv(xi )∞ i=p+1 for some p ∈ N. The following theorem, motivated by [25], will play a significant role in the sequel. Theorem 2.1. Let (xn ) be a bounded sequence in a Banach space X. For every ε > 0 there exists a sequence (yn ) of scc for (xn ) such that if u1 , u2 and v1 , v2 are any couples of scc for (yn ), then |ku1 − u2 k − kv1 − v2 k| ≤ ε. T∞ Proof. Let An = Conv(xi )∞ i=n for n ∈ N. Assume first that n=1 An 6= ∅. Then we can find a convergent sequence of scc for (xn ) and, by ignoring a finite T∞ number of terms if necessary, we get a sequence as required. Suppose now that n=1 An = ∅. WEAK NONCOMPACTNESS AND INTERPOLATION 3 Reasoning similar to that in the proof of the corollary of Theorem 2 [25] (see also [26]) gives a constant d > 0 such that for any ε > 0 there exists (yn ) of scc for (xn ) such that d − ε ≤ ku1 − u2 k ≤ d for any couple u1 , u2 of scc for (yn ). Clearly, (yn ) fulfills the assertion of the theorem.  Now, following [6], we introduce an axiomatic approach to the notion of a measure of weak noncompactness. Let µ be a real-valued function defined on the family of all bounded and nonempty subsets of a Banach space X. We call µ a measure of weak noncompactness on X, if the following conditions are satisfied for any subsets A, B and c ∈ R: (1) µ(A) = 0 if and only if A is a relatively weakly compact set; (2) if A ⊂ B, then µ(A) ≤ µ(B); (3) µ(Conv A) = µ(A); (4) µ(A ∪ B) = max{µ(A), µ(B)}; (5) µ(A + B) ≤ µ(A) + µ(B); (6) µ(cA) = |c| µ(A). Definition 2.2. We introduce the function γ defined for every nonempty and bounded set A ⊂ X by the formula γ(A) = sup{csep(xn ) : (xn ) ⊂ conv A}, where csep(xn ) = inf{ky1 − y2 k : y1 , y2 is a couple of scc for (xn )}. Theorem 2.3. γ is a measure of weak noncompactness on X. Proof. Conditions (2),(3) and (6) are easy to check. Implications in (1) for γ are consequences of the James theorem [19] and Mazur theorem [24] respectively. To prove (5), let us take bounded A, B ⊂ X and a sequence (xnP ) ⊂ conv(A + mn n B). For every n ∈ N there exist nonnegative λn1 , . . . , λnmn , where i=1 λi = 1, P m n n n n n and elements u ∈ A, v ∈ B, 1 ≤ i ≤ m such that x = λ (u + vin ). n n i i i i i=1 Pmn n n Pmn n n Let yn = i=1 λi ui , zn = i=1 λi vi and fix ε > 0. By Theorem 2.1 there exists a sequence (yn′ ) of scc for (yn ) such that |kw1 − w2 k − kw1′ − w2′ k| ≤ ε for ′ ′ ′ term in (yn′ ) takes the form any couples Pnk+1 w1 , wk2 and w1 , w2k of scc for (yn ). Each P nk+1 ′ k ′ yk = i=nk +1 µi yi , where µi are nonnegative and i=nk +1 µi = 1. Set zk = P nk+1 k ′′ ′ i=nk +1 µi zi . Applying again Theorem 2.1 we get a sequence (zn ) of scc for (zn ) ′′ ′′ ′′ such that |kv1 − v2 k − kz1 − z2 k| ≤ ε for any couple v1 , v2 of scc for (zn ). For such obtained (zn′′ ) we can construct a sequence (yn′′ ) of scc for (yn′ ) in the similar way as we have constructed (zn′ ) for a given (yn′ ). Then ky1′′ − y2′′ k ≤ csep(yn′′ ) + ε and kz1′′ − z2′′ k ≤ csep(zn′′ ) + ε. Let us notice that x′′1 = y1′′ + z1′′ , x′′2 = y2′′ + z2′′ form a couple of scc for (xn ) and therefore csep(xn ) ≤ kx′′1 − x′′2 k ≤ ky1′′ − y2′′ k + kz1′′ − z2′′ k ≤ csep(yn′′ ) + csep(zn′′ ) + 2ε ≤ γ(A) + γ(B) + 2ε. An arbitrary choice of ε yields csep(xn ) ≤ γ(A) + γ(B). Thus γ(A + B) ≤ γ(A) + γ(B). To obtain (4), first we see from (2) that max{γ(A), γ(B)} ≤ γ(A ∪ B). For the opposite inequality let us choose a sequence (xn ) in conv(A ∪ B). Each xn takes the form xn = tn yn + (1 − tn )zn for some tn ∈ [0, 1], yn ∈ conv A and zn ∈ conv B. 4 A. KRYCZKA, S. PRUS, AND M. SZCZEPANIK There exists a subsequence (tnk ) convergent to some t ∈ [0, 1]. The sets A, B are bounded, so M = sup{kxk : x ∈ A ∪ B} < ∞ and for arbitrarily fixed ε > 0 we can assume that |tnk − t| ≤ Mε+1 for all k. Then k(tnk − t)ynk k ≤ Mε+1 kynk k ≤ ε and similarly k(1 − tnk )znk − (1 − t)znk k ≤ ε. It follows that kxnk − uk k ≤ 2ε, where uk = tynk + (1 − t)znk . Since (uk ) is a sequence in t conv A + (1 − t) conv B, we have csep(uk ) ≤ γ(t conv A + (1 − t) conv B) ≤ tγ(A) + (1 − t)γ(B) ≤ max{γ(A), γ(B)}. But csep(xnk ) ≤ csep(uk ) + 4ε, which shows that csep(xn ) ≤ csep(xnk ) ≤ max{γ(A), γ(B)} + 4ε. Finally, γ(A ∪ B) ≤ max{γ(A), γ(B)}.  In the next two results we establish alternative formulae for the measure γ. Theorem 2.4. Let A be a nonempty and bounded subset of a Banach space X. Then (2.1) γ(A) = sup dist(x∗∗ , conv(xn )) , where the supremum is taken over all sequences (xn ) in conv A and all w∗-cluster points x∗∗ ∈ X ∗∗ of a sequence (xn ). Proof. Let us denote by γ ′ (A) the right-hand side of formula (2.1). In order to show that γ ′ (A) ≥ γ(A) we deduce likewise in the proof of Lemma 2.1 in [26]. Indeed, for ε > 0 let us find a sequence (xn ) in conv A such that γ(A) − ε ≤ csep(xn ) and fix x ∈ conv(xn ). There exists m ∈ N such that if y ∈ conv(xi )∞ i=m , then x, y is a couple of scc for (xn ) and therefore kx − yk ≥ csep(xn ) ≥ γ(A) − ε. Applying a separation theorem we can find a functional x∗ ∈ X ∗ such that kx∗ k ≤ 1 and ∗∗ x∗ (y − x) ≥ γ(A) − ε for all y ∈ conv(xi )∞ ∈ X ∗∗ be a w∗-cluster i=m . Let x ∗∗ ∗ ∗ point of (xn ). Then x (x ) = limk→∞ x (xnk ) for some subsequence (xnk ) and consequently kx∗∗ − xk ≥ (x∗∗ − x)(x∗ ) = limk→∞ x∗ (xnk − x) ≥ γ(A) − ε. This gives dist(x∗∗ , conv(xn )) ≥ γ(A) − ε and finally, γ ′ (A) ≥ γ(A). The proof of the opposite inequality is a modification of reasoning in [19] (see also [17]). For each ε > 0 there exists a sequence (xn ) in conv A and its w∗-cluster point x∗∗ ∈ X ∗∗ such that dist(x∗∗ , conv(xn )) ≥ γ ′ (A) − ε. By a separation theorem we obtain a functional x∗∗∗ ∈ X ∗∗∗ such that kx∗∗∗ k ≤ 1 and x∗∗∗ (x∗∗ − x) ≥ γ ′ (A)−ε for all x ∈ conv(xn ). We now choose by induction a sequence of functionals (x∗k ) ⊂ X ∗ and a subsequence (xnk ) with the following properties: kx∗k k ≤ 1 + ε for all k ∈ N, x∗k (xni ) ≥ x∗∗∗ (x∗∗ ) − ε if k ≤ i and x∗k (xni ) = x∗∗∗ (xni ) if i < k. To construct these sequences, we apply the principle of local reflexivity [17, p.33]. By this principle, there exists x∗1 ∈ X ∗ such that kx∗1 k ≤ 1 + ε and x∗∗ (x∗1 ) = x∗∗∗ (x∗∗ ). Since x∗∗ is the w∗-cluster point of (xn ) we can find n1 ∈ N satisfying |x∗1 (xn1 ) − x∗∗ (x∗1 )| ≤ ε. Let us assume now, that we have obtained required x∗1 , . . . , x∗k−1 and xn1 , . . . , xnk−1 . There exists x∗k ∈ X ∗ such that kx∗k k ≤ 1 + ε, x∗k (xni ) = x∗∗∗ (xni ) for i = 1, . . . , k − 1 and x∗∗ (x∗k ) = x∗∗∗ (x∗∗ ). We choose a number nk > nk−1 to satisfy |x∗i (xnk ) − x∗∗ (x∗i )| ≤ ε for i = 1, . . . , k. Let us consider now a couple u, v of scc for (xni ). There exists k > 1 such that k−1 ∗ ∗∗∗ (u) and x∗k (v) ≥ , v ∈ conv(xni )∞ u ∈ conv(xni )i=1 i=k . Of course xk (u) = x ∗∗∗ ∗∗ ∗∗∗ ∗∗ ∗ x (x )−ε, and hence (1+ε) ku − vk ≥ xk (v −u) ≥ x (x −u)−ε ≥ γ ′ (A)−2ε. WEAK NONCOMPACTNESS AND INTERPOLATION ′ , which gives γ(A) ≥ From this it follows that csep(xni ) ≥ γ (A)−2ε 1+ε ε → 0 we obtain γ(A) ≥ γ ′ (A), and the proof is complete. 5 γ ′ (A)−2ε . 1+ε Letting  The measure γ is equal to the function measuring the deviation from relative weak compactness based on the classical double-limit criterion. The function of this type was considered also in [4]. Theorem 2.5. Let A be a nonempty and bounded subset of a Banach space X. Then (2.2) γ(A) = sup{lim lim Fn (xk ) − lim lim Fn (xk ) : n k k n (xk ) ⊂ conv A, (Fn ) ⊂ BX ∗ and the limits exist}. Proof. Let us denote by e(A) the right-hand side of formula (2.2). From the proof of Theorem 2.4 it follows that there exists a such that for every ε > 0 one can find sequences (xk ) ⊂ conv A and (Fn ) ⊂ BX ∗ satisfying the following: Fn (xk ) ≥ a − ε for n ≤ k and Fn (xk ) ≤ a − γ(A) + ε for n > k. Passing to a subsequence, if necessary, we can assume that all the limits α1 = limn limk Fn (xk ) and α2 = limk limn Fn (xk ) exist. Of course, γ(A) − 2ε ≤ α1 − α2 . Hence γ(A) − 2ε ≤ e(A) and γ(A) ≤ e(A). Let (xk ) ⊂ conv A and (Fn ) ⊂ BX ∗ be any sequences for which the limits β1 = limn limk Fn (xk ) and β2 = limk limn Fn (xk ) exist. By Theorem 2.1 for arbitrary ε > 0 we can find a sequence (zk ) of scc for (xk ) such that kzi − zj k − csep(zk ) ≤ ε for every i 6= j. Then β1 = limn limk Fn (zk ) and β2 = limk limn Fn (zk ). Therefore β1 − β2 ≤ lim inf j lim inf i kzi − zj k. Consequently, we get β1 − β2 ≤ csep(zk ) + ε ≤ γ(A) + ε. Hence γ(A) + ε ≥ e(A) and finally γ(A) ≥ e(A).  2.2. γ and De Blasi’s measure ω. The axiomatic approach to a measure of weak noncompactness, presented in subsection 2.1, was preceded by a definition of the measure ω introduced by F.S. De Blasi (see [16] and the references given there). For a nonempty and bounded subset A of a Banach space X let ω(A) = inf{t > 0 : A ⊂ C + tBX and C ⊂ X is a weakly compact set}. Theorem 2.5 and a result of K. Astala and H.-O. Tylli (see [4]) show that, in general, the measures γ and ω are not equivalent. Certainly, if X is reflexive, then BX is weakly compact and ω(BX ) = 0. Otherwise it is shown [16] that ω(BX ) = 1. Similarly, for reflexive spaces γ(BX ) = 0. The nonreflexive case for γ differs from the one for ω. Example 2.6. Consider the vectors en = (0, . . . , 0, 1, 0, . . .) ∈ l1 , n ∈ N, where 1 occupies the n-th place. It is evident that csep(en ) = 2 and therefore γ(Bl1 ) = 2. Example 2.7. The case of c0 is more complicated. From the next theorem we will see at once that γ(Bc0 ) = 1. Theorem 2.8. Let A be a nonempty and bounded subset in c0 . Then (2.3) γ(A) = sup dist(x∗∗ , c0 ) , where the supremum is taken over all sequences (xn ) in conv A and all w∗-cluster points x∗∗ ∈ c∗∗ 0 of (xn ). 6 A. KRYCZKA, S. PRUS, AND M. SZCZEPANIK Proof. Let us denote by γ ′ (A) the right-hand side of formula (2.3). In view of Theorem 2.4 it is enough to prove that γ(A) ≤ γ ′ (A). Choose M such that A ⊂ M Bc0 and fix a sequence (xn ) ⊂ conv A. For each w∗-cluster point x∗∗ = (x∗∗ (k)) of (xn ) there exists a subsequence (xni ) such that x∗∗ = w∗- limi→∞ xni . Let q = dist (x∗∗ , c0 ) = lim supk→∞ |x∗∗ (k)| and yi = xni for i ∈ N. Fix ε > 0 and N N +1 N ∈ N. Let us choose a subsequence (yni )i=1 and an increasing sequence (ki )i=1 of ∗∗ natural numbers such that |x (k)| < q + ε for k > k1 , and satisfying the following conditions: |yni (k) − x∗∗ (k)| < ε for k ≤ ki and |yni (k)| < ε for k > ki+1 , PN i = 1, . . . , N . Let us define x = N1 i=1 yni ∈ conv(xn ). Then for k > k1 we have |x (k) − x∗∗ (k)| < |x∗∗ (k)| + ε + M M < q + 2ε + N N and for k ≤ k1 |x (k) − x∗∗ (k)| < ε. Both the inequalities and an arbitrary choice of ε and N yields dist (x∗∗ , conv(xn )) ≤ q and hence γ (A) ≤ γ ′ (A).  Theorem 2.9. The measures γ and ω are equal in c0 . Proof. Having q ≥ 0, we set rq (α) =  0 α α − q |α| if |α| ≤ q if |α| > q. Next, let Rq : c0 → c0 be given by the formula Rq x = (rq (x (k))), where x = (x(k)). We take a nonempty bounded set A ⊂ c0 and put q = γ(A). We will show that γ (Rq (A)) = 0. Let (xn ) ⊂ A and suppose that there exists the limit ∗∗ w∗- limn→∞ xn = x∗∗ ∈ c∗∗ 0 . By Theorem 2.8 lim supk→∞ |x (k)| ≤ q. Therefore     ∗∗ ∗∗ ∗ lim sup w - lim Rq xn (k) = lim sup |rq (x (k))| = rq lim sup |x (k)| = 0. k→∞ n→∞ k→∞ k→∞ In view of Theorem 2.8 we obtain γ(Rq (A)) = 0. It follows that Rq (A) is relatively weakly compact. Since A ⊂ Rq (A) + qBc0 , we get ω (A) ≤ q = γ (A).  On the other hand, we have γ(A) ≤ γ(Bc0 )ω(A) = ω(A) (see [7]). 3. Real interpolation of operators 3.1. Measure of weak noncompactness for operators. By L(X, Y ) we denote the space of all linear and bounded operators T : X → Y between Banach spaces X and Y . Using the measure γ, introduced in the previous section, we can define the measure of weak noncompactness for every operator T ∈ L(X, Y ). This measure provides in addition a seminorm in L(X, Y ). Definition 3.1. For every T ∈ L(X, Y ) the number Γ(T ) = γ(T (BX )) will be called the measure of weak noncompactness of the operator T . Let X be a Banach space. We shall denote by lp (X), 1 < p < ∞ the Banach space of all sequences x = (x(i)) such that x(i) ∈ X for all i ∈ N and kxklp (X) = P∞ p 1/p ( i=1 kx(i)k ) is finite. A standard verification shows that we can identify ∗ (lp (X)) withPlq (X ∗ ), where 1/p + 1/q = 1, and for each φ = (φ(i)) ∈ (lp (X))∗ we ∞ have φ(x) = i=1 φ(i)(x(i)) for all x ∈ lp (X) (see [21]). If X, Y are Banach spaces WEAK NONCOMPACTNESS AND INTERPOLATION 7 and T ∈ L(X, Y ), then the operator Te : lp (X) → lp (Y ) given by Tex = (T x(i)) is also bounded and kTek = kT k. To deal with the measure γ, ultrafilters will be used in several cases. For more details concerning filters we refer the reader for instance to [2]. We recall two important facts. Lemma 3.2. Let U be an ultrafilter on a nonempty set I and I1 ∪ I2 = I. Then I1 or I2 belongs to U. Lemma 3.3. An ultrafilter U is free if and only if U does not contain a finite set. Let τ be a Hausdorff topology in a space E and U be an ultrafilter on the set of positive integers N. An element x ∈ E is said to be the limit over U of a sequence (xn ) in E, if {n ∈ N : xn ∈ V } ∈ U for every neighbourhood V of x. Then we write x = τ -limU xn or simply x = limU xn . Let us recall that if E is compact, then limU xn exists for each sequence (xn ) in E. The next lemma, roughly speaking, can stand for a passing to a subsequence. In the light of Lemma 3.3, the proof is straightforward. Lemma 3.4. Let U be a free ultrafilter on N and N1 ∈ U. If f : N1 → N is the bijection given by f (nk ) = k, then U1 = {f (N1 ∩ A) : A ∈ U} is a free ultrafilter on N. Moreover, if limU xn = x in a topological space X, then limU1 xnk = x as well. Lemma 3.5. Let Y be a Banach space and 1 < p < ∞. If y = (y(i)), yn = (yn (i)) ∈ lp (Y ∗∗ ) for all n ∈ N and y = w∗- limU yn over some free ultrafilter U on N, then y(i) = w∗- limU yn (i) for each i ∈ N. Proof. Let us first recall that if f : X → Y is continuous function between topological spaces, (xi )i∈I ⊂ X and x = limU ′ xi over a free ultrafilter U ′ on a set of indices I, then f (x) = limU ′ f (xi ). Fix i ∈ N and take v = (0, . . . , 0, v(i), 0, . . .) ∈ lq (Y ∗ ). The functional fv given by the formula fv (z) = z(v) = z(i)(v(i)) for z ∈ lp (Y ∗∗ ) is w∗-continuous. Consequently, fv (yn ) = yn (i)(v(i)) and y(i)(v(i)) = fv (y) = w∗- limU yn (i)(v(i)) for every v(i) ∈ Y ∗ which is equivalent to y(i) = w∗- limU yn (i).  Theorem 3.6. Let X, Y be Banach spaces and 1 < p < ∞. If T ∈ L(X, Y ) and Te ∈ L(lp (X), lp (Y )) is defined by Tex = (T x(i)) for x = (x(i)) ∈ lp (X), then Γ(T ) = Γ(Te). Proof. Since T = Te|{(x, 0, 0, . . .)}, we see that Γ(T ) ≤ Γ(Te). For the proof of Γ(T ) ≥ Γ(Te), assume that Γ(Te) > 0, otherwise at once Γ(T ) = Γ(Te). Fix 0 < ε < Γ(Te) and let Γ1 = Γ(Te) − ε. There exists (xn ) ⊂ Blp (X) such that for yn = Texn we have 0 < Γ1 ≤ dist(y, conv(yn )) ≤ dist(y, (yn )), where y ∈ lp (Y ∗∗ ) is a w∗-cluster point of the sequence (yn ) and therefore y = w∗- limU yn over some free ultrafilter U in N. By a separation theorem there exists φ ∈ lq (Y ∗∗∗ ) such P∞ that kφk ≤q 1 and φ(z) ≥ Γ1 for all z ∈ y −conv(yn ). Now, fix m ∈ N satisfying i=m+1 kφ(i)k < εq . The boundedness of y − conv(yn ) gives a constant c > 0 such that !1/q !1/p m ∞ ∞ X X X q p Γ1 ≤ φ(i)(y(i) − yn (i)) + kφ(i)k ky(i) − yn (i)k i=1 ≤ m X i=1 i=m+1 φ(i)(y(i) − yn (i)) + εc i=m+1 8 A. KRYCZKA, S. PRUS, AND M. SZCZEPANIK for every n ∈ N, where, by Lemma 3.5, y(i) = w∗- limU yn (i) for all i. Writing I = {1 ≤ i ≤ m : φ(i) 6= 0} and ψ(i) = φ(i) for i ∈ I, kφ(i)k αi = lim kxn (i)k , U v(i) = y(i) ε , αi + m vn (i) = yn (i) ε , αi + m by Hölder’s and Minkowski’s inequalities we obtain X ε kφ(i)k (αi + )ψ(i) (v(i) − vn (i)) Γ1 − εc ≤ m i∈I !1/p ! 1/q m m X X ε p q max ψ(i) (v(i) − vn (i)) (αi + ) ≤ kφ(i)k i∈I m i=1 i=1 ≤ (1 + εm1/p−1 ) max ψ(i) (v(i) − vn (i)) . i∈I We set Γ2 = (Γ1 − εc)(1 + εm (3.1) 1/p−1 −1 ) . Then Γ2 ≤ max ψ(i) (v(i) − vn (i)) i∈I for all n ∈ N. For each 1 ≤ i ≤ m let Ni denote the set of all natural S numbers n for which the maximum in (3.1) is attained for i. It is clear that 1≤i≤m Ni = N and, by Lemma 3.2, Nj ∈ U for some j. Let us apply Lemma 3.4 for the set ε } = {nk } ∈ U and change U for a free ultrafilter Nj ∩ {n ∈ N : |kxn (j)k − αj | < m U1 given by this lemma. It follows that Γ2 ≤ ψ(j) (v(j) − vnk (j)) for k ∈ N. Considering convex combinations of the obtained sequence yields Γ2 ≤ dist (v(j), conv(vnk (j))) . But (vnk (j)) ⊂ T (BX ) and v(j) = w∗- limU1 vnk (j), which gives Γ2 ≤ Γ(T ). Letting ε → 0 we conclude that Γ(Te) ≤ Γ(T ) and this finishes the proof.  Remark 3.7. We can consider a space lp (X) of sequences indexed by the set of all integers Z. The same properties as stated at the beginning of this section, Lemma 3.5 and Theorem 3.6 also hold in that case. 3.2. Real interpolation. We shall say that Banach spaces A0 and A1 are compatible if they are continuously embedded in a Hausdorff topological vector space X. Then A0 ∩ A1 and A0 + A1 with norms kakA0 ∩A1 = max{kakA0 , kakA1 } and kakA0 +A1 = inf a=a0 +a1 {ka0 kA0 + ka1 kA1 } respectively, are also Banach spaces. Let A = (A0 , A1 ) denote a couple of compatible Banach spaces A0 and A1 . A Banach space A is said to be an intermediate space with respect to A if A0 ∩ A1 ⊂ A ⊂ A0 + A1 and both inclusions are continuous. Let A = (A0 , A1 ) and B = (B0 , B1 ) be two couples of compatible Banach spaces and T a linear operator from A0 + A1 into B0 +B1 . We shall write T : A → B for brevity, if T ∈ L(A0 , B0 ) and T ∈ L(A1 , B1 ), viewed as the restrictions of T . If A and B are intermediate spaces with respect to A and B respectively, and T : A → B implies T ∈ L(A, B), then A and B are said to be interpolation spaces with respect to A and B. WEAK NONCOMPACTNESS AND INTERPOLATION 9 In the sequel, we restrict our considerations to one of the equivalent constructions of so called real interpolation spaces, i.e. a discrete method introduced by J.-L. Lions and J. Peetre [21]. Let 0 < θ < 1, 1 < p < ∞ and for each a ∈ A0 + A1 kakθ,p = inf max{k(2iθ a0 (i))klp (A0 ) , k(2i(θ−1) a1 (i))klp (A1 ) }, where the infimum is taken over all (a0 (i)) ⊂ A0 , (a1 (i)) ⊂ A1 such that a0 (i) + a1 (i) = a for all i ∈ Z. Then Aθ,p = {a ∈ A0 + A1 : kakθ,p < ∞} is an intermediate space with respect to A and moreover one can show [21] that for a ∈ Aθ,p (3.2) kakθ,p ≤ 2θ(1−θ) inf a0 (i)+a1 (i)=a i(θ−1) k(2iθ a0 (i))k1−θ a1 (i))kθlp (A1 ) . lp (A0 ) k(2 The spaces Aθ,p and Bθ,p , obtained by this method, are interpolation spaces with respect to A and B. Furthermore, for every T : A → B 1−θ kT kθ,p ≤ 2θ(1−θ) kT k0 θ kT k1 , where kT kθ,p and kT ki , i = 0, 1 are norms of operators T : Aθ,p → Bθ,p and T : Ai → Bi , i = 0, 1 respectively. We prove an analogous inequality for our measure of weak noncompactness for operators. Note that a similar result for De Blasi’s measure does not hold (see [12]). Theorem 3.8. Let Aθ,p and Bθ,p be interpolation spaces with respect to A = (A0 , A1 ) and B = (B0 , B1 ) obtained by the real method described above for some 0 < θ < 1 and 1 < p < ∞. Then for every T : A → B Γθ,p (T ) ≤ 2θ(1−θ) Γ0 (T )1−θ Γ1 (T )θ , where Γθ,p and Γi , i = 0, 1 are measures of weak noncompactness for operators T : Aθ,p → Bθ,p and T : Ai → Bi , i = 0, 1 respectively. Proof. Fix ε > 0 and a sequence (an ) ⊂ BAθ,p . For each an there exist (2iθ a0n (i))i∈Z ∈ Blp (A0 ) and (2i(θ−1) a1n (i))i∈Z ∈ Blp (A1 ) such that a0n (i) + a1n (i) = an for all i ∈ Z. Let yn = (2iθ T a0n (i))i∈Z , zn = (2i(θ−1) T a1n (i))i∈Z and bn = T an for every n ∈ N. By a similar method as in the proof of condition (5) of Theorem 2.3, we can obtain sequences (yn′′ ), (zn′′ ) of scc forP (yn ), (zn ) respectively. They satisfy therefore the assertion of Theorem 2.1, Pnk+1 nk+1 ′′ k k yk′′ = j=n λ = n < n2 < n3 < . . . y and z = j j j=nk +1 λj zj for some 0 k k +1 Pnk+11 k k and some nonnegative coefficients λnk +1 , . . . , λnk+1 with j=nk +1 λkj = 1. We set Pnk+1 b′′k = j=n λkj bj for k = 1, 2. Then k +1 1−θ θ csep(bn ) ≤ kb′′1 − b′′2 kθ,p ≤ 2θ(1−θ) ky1′′ − y2′′ klp (B0 ) kz1′′ − z2′′ klp (B1 ) , the last inequality being a consequence of (3.2). But ky1′′ − y2′′ klp (B0 ) ≤ csep(yn′′ ) + f0 (Bl (A ) ) and zn′′ ∈ ε and kz1′′ − z2′′ klp (B1 ) ≤ csep(zn′′ ) + ε. Moreover, yn′′ ∈ T p 0 f1 (Bl (A ) ) for all n ∈ N, where Tei : lp (Ai ) → lp (Bi ), i = 0, 1 is defined as in T p 1 Theorem 3.6. Hence f0 ) + ε)1−θ (Γ(T f1 ) + ε)θ . csep(bn ) ≤ 2θ(1−θ) (Γ(T Finally, Theorem 3.6 with Remark 3.7 and an arbitrary choice of ε and (an ) lead to the desired conclusion.  10 A. KRYCZKA, S. PRUS, AND M. 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Mazur, ‘Über konvexe Mengen in linearen normierten Räumen’, Studia Math. 4 (1933), 70–84. [25] D.P. Milman and V.D. Milman, ‘The geometry of imbeddings with empty intersection. The structure of the unit sphere in a non-reflexive space’, Mat. Sbornik 66 (1965), 109–118 (Russian). [26] V.D. Milman, ‘Geometric theory of Banach spaces. II. Geometry of the unit ball’, Uspehi Mat. Nauk 26 (1971), 73–149 (Russian). English translation: Russian Math. Surv. 26 (1971), 79– 163. [27] M.F. Teixeira and D.E. Edmunds, ‘Interpolation theory and measures of non-compactness’, Math. Nachr. 104 (1981), 129–135. Institute of Mathematics, Maria Curie-Sklodowska University, 20-031 Lublin, Poland E-mail address: akryczka@golem.umcs.lublin.pl Institute of Mathematics, Maria Curie-Sklodowska University, 20-031 Lublin, Poland E-mail address: bsprus@golem.umcs.lublin.pl Institute of Mathematics, Maria Curie-Sklodowska University, 20-031 Lublin, Poland E-mail address: szczepan@golem.umcs.lublin.pl