Mathematical Proceedings of the Royal Irish Academy, 2011
ABSTRACT We study the range-kernel weak orthogonality of certain elementary operators induced by ... more ABSTRACT We study the range-kernel weak orthogonality of certain elementary operators induced by non-normal operators, with respect to the usual operator norm and the von-Neumann-Schatten p-norm (1≤p<∞).
In this paper we characterize Hermitian operators defined on Hilbert space. Using this result we ... more In this paper we characterize Hermitian operators defined on Hilbert space. Using this result we establish several new characterizations to () class operators. Further, we apply these results to investigate on the relation between this class and other usual classes of operators. Some applications are also given.
Introducing the concept of the normalized duality mapping on normed linear space and normed algeb... more Introducing the concept of the normalized duality mapping on normed linear space and normed algebra, we extend the usual definitions of the numerical range from one operator to two operators. In this note we study the convexity of these types of numerical ranges in normed algebras and linear spaces. We establish some Birkhoff-James orthogonality results in terms of the algebra numerical range V (T)A which generalize those given by J.P. William and J.P. Stamplfli. Finally, we give a positive answer of the Mathieu's question. .
We generalize the notion of Fuglede-Putnam's property to general ??Banach algebra in the sens... more We generalize the notion of Fuglede-Putnam's property to general ??Banach algebra in the sense of Fuglede operator and study the elementary operator of length ? 2 in the context of this property.
The characterization of the points in C p : 1 ≤ p < ∞ ( ℋ ) {C}_{p{:}_{1\le p\lt \infty }}\lef... more The characterization of the points in C p : 1 ≤ p < ∞ ( ℋ ) {C}_{p{:}_{1\le p\lt \infty }}\left({\mathcal{ {\mathcal H} }}) , the Von Neuman-Schatten p-classes, that are orthogonal to the range of elementary operators has been done for certain kinds of elementary operators. In this paper, we shall study this problem of characterization on an abstract reflexive, smooth and strictly convex Banach space for arbitrary operator. As an application, we consider other kinds of elementary operators defined on the spaces C p : 1 ≤ p < ∞ ( ℋ ) {C}_{p{:}_{1\le p\lt \infty }}\left({\mathcal{ {\mathcal H} }}) , and finally, we give a counterexample to Mecheri’s result given in this context.
The characterization of the points in C1ℋ, the trace class operators, that are orthogonal to the ... more The characterization of the points in C1ℋ, the trace class operators, that are orthogonal to the range of elementary operators has been carried out for certain kinds of elementary operators by many authors in the smooth case. In this note, we study that the characterization is a problem in nonsmoothness case for general elementary operators, and we give a counter example to S. Mecheri and M. Bounkhel results.
We study the range-kernel weak orthogonality of certain elementary operators induced by non-norma... more We study the range-kernel weak orthogonality of certain elementary operators induced by non-normal operators, with respect to usual operator norm and the Von Newmann-Schatten p p -norm ( 1 ≤ p < ∞ ) \left(1\le p\lt \infty ) .
T ∈ B ( H ) is said to be ( n , k ) -quasi-∗-paranormal operator if, for non-negative integers k ... more T ∈ B ( H ) is said to be ( n , k ) -quasi-∗-paranormal operator if, for non-negative integers k and n, ∥ T ∗ ( T k x ) ∥ ( 1 + n ) ≤ ∥ T ( 1 + n ) ( T k x ) ∥ ∥ T k x ∥ n ; for all x ∈ H . In this paper, the asymmetric Putnam-Fuglede theorem for the pair ( A , B ) of power-bounded operators is proved when (i) A and B ∗ are n-∗-paranormal operators (ii) A is a ( n , k ) -quasi-∗-paranormal operator with reduced kernel and B ∗ is n-∗-paranormal operator. The class of ( n , k ) -quasi-∗-paranormal operators properly contains the classes of n-∗-paranormal operators, ( 1 , k ) -quasi-∗-paranormal operators and k-quasi-∗-class A operators. As a consequence, it is showed that if T is a completely non-normal ( n , k ) -quasi-∗-paranormal operator for k = 0 , 1 such that the defect operator D T is Hilbert-Schmidt class, then T ∈ C 10 .
Mathematical Proceedings of the Royal Irish Academy, 2011
ABSTRACT We study the range-kernel weak orthogonality of certain elementary operators induced by ... more ABSTRACT We study the range-kernel weak orthogonality of certain elementary operators induced by non-normal operators, with respect to the usual operator norm and the von-Neumann-Schatten p-norm (1≤p&lt;∞).
In this paper we characterize Hermitian operators defined on Hilbert space. Using this result we ... more In this paper we characterize Hermitian operators defined on Hilbert space. Using this result we establish several new characterizations to () class operators. Further, we apply these results to investigate on the relation between this class and other usual classes of operators. Some applications are also given.
Introducing the concept of the normalized duality mapping on normed linear space and normed algeb... more Introducing the concept of the normalized duality mapping on normed linear space and normed algebra, we extend the usual definitions of the numerical range from one operator to two operators. In this note we study the convexity of these types of numerical ranges in normed algebras and linear spaces. We establish some Birkhoff-James orthogonality results in terms of the algebra numerical range V (T)A which generalize those given by J.P. William and J.P. Stamplfli. Finally, we give a positive answer of the Mathieu's question. .
We generalize the notion of Fuglede-Putnam's property to general ??Banach algebra in the sens... more We generalize the notion of Fuglede-Putnam's property to general ??Banach algebra in the sense of Fuglede operator and study the elementary operator of length ? 2 in the context of this property.
The characterization of the points in C p : 1 ≤ p < ∞ ( ℋ ) {C}_{p{:}_{1\le p\lt \infty }}\lef... more The characterization of the points in C p : 1 ≤ p < ∞ ( ℋ ) {C}_{p{:}_{1\le p\lt \infty }}\left({\mathcal{ {\mathcal H} }}) , the Von Neuman-Schatten p-classes, that are orthogonal to the range of elementary operators has been done for certain kinds of elementary operators. In this paper, we shall study this problem of characterization on an abstract reflexive, smooth and strictly convex Banach space for arbitrary operator. As an application, we consider other kinds of elementary operators defined on the spaces C p : 1 ≤ p < ∞ ( ℋ ) {C}_{p{:}_{1\le p\lt \infty }}\left({\mathcal{ {\mathcal H} }}) , and finally, we give a counterexample to Mecheri’s result given in this context.
The characterization of the points in C1ℋ, the trace class operators, that are orthogonal to the ... more The characterization of the points in C1ℋ, the trace class operators, that are orthogonal to the range of elementary operators has been carried out for certain kinds of elementary operators by many authors in the smooth case. In this note, we study that the characterization is a problem in nonsmoothness case for general elementary operators, and we give a counter example to S. Mecheri and M. Bounkhel results.
We study the range-kernel weak orthogonality of certain elementary operators induced by non-norma... more We study the range-kernel weak orthogonality of certain elementary operators induced by non-normal operators, with respect to usual operator norm and the Von Newmann-Schatten p p -norm ( 1 ≤ p < ∞ ) \left(1\le p\lt \infty ) .
T ∈ B ( H ) is said to be ( n , k ) -quasi-∗-paranormal operator if, for non-negative integers k ... more T ∈ B ( H ) is said to be ( n , k ) -quasi-∗-paranormal operator if, for non-negative integers k and n, ∥ T ∗ ( T k x ) ∥ ( 1 + n ) ≤ ∥ T ( 1 + n ) ( T k x ) ∥ ∥ T k x ∥ n ; for all x ∈ H . In this paper, the asymmetric Putnam-Fuglede theorem for the pair ( A , B ) of power-bounded operators is proved when (i) A and B ∗ are n-∗-paranormal operators (ii) A is a ( n , k ) -quasi-∗-paranormal operator with reduced kernel and B ∗ is n-∗-paranormal operator. The class of ( n , k ) -quasi-∗-paranormal operators properly contains the classes of n-∗-paranormal operators, ( 1 , k ) -quasi-∗-paranormal operators and k-quasi-∗-class A operators. As a consequence, it is showed that if T is a completely non-normal ( n , k ) -quasi-∗-paranormal operator for k = 0 , 1 such that the defect operator D T is Hilbert-Schmidt class, then T ∈ C 10 .
Uploads
Papers by Abdelkader Segres