Birkhoff-James Orthogonality and Its Local Symmetry in Some Sequence Spaces

Abstract

We study Birkhoff-James orthogonality and its local symmetry in some sequence spaces namely ℓp, for 1 ≤ p ≤ ∞, p = 2, c, c0 and c00. Using the characterization of the local symmetry of Birkhoff-James orthogonality, we characterize isometries of each of these spaces onto itself and obtain the Banach-Lamperti theorem for onto operators on the sequence spaces.

Birkhoff-James Orthogonality and Its Pointwise Symmetry in Some Function Spaces

“…Note that by the term pointwise symmetry of Birkhoff-James orthogonality, we refer to the left-symmetric and the rightsymmetric points of a given normed linear space. The left-symmetric and the right-symmetric points of ℓ p spaces where 1 ≤ p ≤ ∞, p = 2, were characterized in [3]. Here we generalize these results in L p (X) for any measure space X and p ∈ [1, ∞] \ {2}.…”

“…The present article aims to explore Birkhoff-James orthogonality and its pointwise symmetry in some function spaces. We have completed such a study for some well studied sequence spaces, namely ℓ p for 1 ≤ p ≤ ∞, c, c 0 and c 00 in [3]. Here we take the study one step further by doing the same for commutative C * algebras and L p (X) for 1 ≤ p ≤ ∞ and any measure space X.…”

Point-wise Symmetry of Birkhoff-James Orthogonality and Geometry of $\mathbb{B}(\ell_\infty^n,\ell_1^m)$

“…In recent times, Birkhoff-James orthogonality and its pointwise symmetry has been used to understand the geometry of a normed linear space. Characterization of Birkhoff-James orthogonality and its local symmetry has been done for finite-dimensional ℓ p spaces in [6], while that for the sequence spaces ℓ p and the function spaces L p have been done in [4], [5]. In these articles, these characterizations have been used to understand the geometry of the underlying spaces by describing the smooth points and the onto isometries of the spaces.…”