On some geometric properties of operator spaces

Abstract

In this paper we study some geometric properties like parallelism, orthogonality and semi-rotundity in the space of bounded linear operators. We completely characterize parallelism of two compact linear operators between normed linear spaces $\mathbb{X}$ and $\mathbb{Y}$, assuming $\mathbb{X}$ to be reflexive. We also characterize parallelism of two bounded linear operators between normed linear spaces $\mathbb{X}$ and $\mathbb{Y}.$ We investigate parallelism and approximate parallelism in the space of bounded linear operators defined on a Hilbert space. Using the characterization of operator parallelism, we study Birkhoff-James orthogonality in the space of compact linear operators as well as bounded linear operators. Finally, we introduce the concept of semi-rotund points (semi-rotund spaces) which generalizes the notion of exposed points (strictly convex spaces). We further study semi-rotund operators and prove that $\mathbb{B}(\mathbb{X},\mathbb{Y})$ is a semi-rotund space which is not strictly convex, if $\mathbb{X},\mathbb{Y}$ are finite-dimensional Banach spaces and $\mathbb{Y}$ is strictly convex.

17 pages

References in corpus (1)

• Norm-parallelism in the geometry of Hilbert $C^{\ast }$-modules

Cited by in corpus (5)

• Birkhoff-James orthogonality and applications : A survey
• Orthogonality and Numerical radius inequalities of operator matrices
• Orthogonality preserving property for pairs of operators on Hilbert $C^{\ast }$-modules
• Characterization of numerical radius parallelism in $C^{\ast }$-algebras
• Approximate Birkhoff-James orthogonality and smoothness in the space of bounded linear operators