Symmetry of Birkhoff–James orthogonality of operators defined between infinite dimensional Banach spaces

Abstract

We study left symmetric bounded linear operators in the sense of Birkhoff-James orthogonality defined between infinite dimensional Banach spaces. We prove that a bounded linear operator defined between two strictly convex Banach spaces is left symmetric if and only if it is zero operator when the domain space is reflexive and Kadets-Klee. We exhibit a non-zero left symmetric operator when the spaces are not strictly convex. We also study right symmetric bounded linear operators between infinite dimensional Banach spaces.2010 Mathematics Subject Classification. Primary 47L05, Secondary 46B20.

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 and Its Local Symmetry in Some Sequence Spaces

“…We now characterize the local symmetry of Birkhoff-James orthogonality in ℓ p . Now, Proposition 2.1 of [14] states that in a smooth, strictly convex space, a point is left-symmetric if and only if it is right-symmetric. Hence the necessity of the right-symmetric case follows from the left-symmetric case.…”

Birkhoff-James orthogonality and applications : A survey

“…So the extreme points of D[0, 1] in B(H) are precisely those operators which are of unit norm and are also right symmetric. There is also a concept of a left symmetric operator, the study of which can be found in [30,71,79,96].…”