Orthogonality and Parallelism of Operators on Various Banach Spaces

Abstract

We present some properties of orthogonality and relate them with support disjoint and norm inequalities in p−Schatten ideals. In addition, we investigate the problem of characterization of norm-parallelism for bounded linear operators. We consider the characterization of norm-parallelism problem in p−Schatten ideals and locally uniformly convex spaces. Later on, we study the case when an operator is norm-parallel to the identity operator. Finally, we give some equivalence assertions about the norm-parallelism of compact operators. Some applications and generalizations are discussed for certain operators.

Numerical Radius Parallelism of Hilbert Space Operators

“…In the setting of normed linear spaces, two linearly dependent vectors are norm-parallel, but the converse is false in general. To see this consider the vectors (1, 0) and (1,1) in the space C 2 with the max-norm. Notice that the norm-parallelism is symmetric and R-homogenous, but is not transitive that is x y and y z does not imply x z in general; see [15,Example 2.7], unless X is smooth at y; see [11, Theorem 3.1]).…”

“…Some characterizations of the norm-parallelism for operators on various Banach spaces and elements of an arbitrary Hilbert C * -module were given in [1,3,9,11,13,14,15]. and so T ω I.…”

Birkhoff–James orthogonality of operators in semi-Hilbertian spaces and its applications

“…Moreover, the papers [18] and [19] show another ways to obtain the Bhatia-Šemrl theorem. Some other authors studied different aspects of orthogonality of operators on various Banach spaces and elements of an arbitrary Hilbert C * -module; see, for instance, [1,5,7,10,11,15,17,20]. Now, let us introduce the notion of A-Birkhoff-James orthogonality of operators in semi-Hilbertian spaces.…”

Numerical radius orthogonality in $$C^*$$-algebras

“…Let a, b be positive elements of A. Then the following statements are equivalent: Some authors extended the well known result of Bhatia-Šemrl (see [3,5,7,12,15,16,19]).…”

“…Similar investigations have been worked out in compact operators spaces for injective operators (cf. [17,Theorems 5.6,5,7,5.8]).…”