Orthogonality of bounded linear operators on complex Banach spaces

Abstract

We study Birkhoff-James orthogonality of bounded linear operators on complex Banach spaces and obtain a complete characterization of the same. By means of introducing new definitions, we illustrate that it is possible in the complex case, to develop a study of orthogonality of bounded (compact) linear operators, analogous to the real case. Furthermore, earlier operator theoretic characterizations of Birkhoff-James orthogonality in the real case, can be obtained as simple corollaries to our present study. In fact, we obtain more than one equivalent characterizations of Birkhoff-James orthogonality of compact linear operators in the complex case, in order to distinguish the complex case from the real case. We also study the left symmetric linear operators on complex two-dimensional lp spaces. We prove that T is a left symmetric linear operator on ℓ 2 p (C) if and only if T is the zero operator.2010 Mathematics Subject Classification. Primary 46B20, Secondary 47L05. 1 2 KALLOL PAUL, DEBMALYA SAIN, ARPITA MAL AND KALIDAS MANDALLet X, Y be complex Banach spaces. Let B X = {x ∈ X : x ≤ 1} and S X = {x ∈ X : x = 1} be the unit ball and the unit sphere of X, respectively. Let L(X, Y)(K(X, Y)) denote the Banach space of all bounded (compact) linear operators from X to Y, endowed with the usual operator norm. We write L(X, Y) = L(X) and K(X,For any two elements x, y ∈ X, x is said to be B-J orthogonal to y, written asSimilarly, for any two elements T, A ∈ L(X), T is said to be B-J orthogonal to A, written asFor a linear operator T defined on a Banach space X, let M T denote the collection of all unit vectors in X at which T attains norm, i.e,

On symmetry of strong Birkhoff orthogonality in $$B(\mathcal {H},\mathcal {K})$$ and $$K(\mathcal {H},\mathcal {K})$$

“…where H is a complex Hilbert space. Since then, many results on local symmetry of Birkhoff orthogonality have been published, especially, in the fields of Banach (or Hilbert) space operators; see [11,13,21,26] for results on spaces of bounded linear operators between Banach spaces, and [3,7,12,[22][23][24][25]27,28] for important techniques about Birkhoff orthogonality in such spaces. See also [18][19][20] for developments in the setting of operator algebras.…”

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.…”

On Uniqueness of New Orthogonality via 2-HH Norm in Normed Linear Space

“…To study the difference of orthogonality in the complex case in comparison with the real case, Paul et al in 2018 came with a new concept of Birkhoff-James orthogonality by introducing new definitions on complex reflexive Banach spaces and introduced more than one equivalent characterization of Birkhoff-James orthogonality of compact linear operators in the complex case [3]. In 1945, James came with the concept of the Pythagorean and isosceles orthogonalities, which characterize inner product space via their homogeneity and additivity [4].…”