## Banach Journal of Mathematical Analysis

### On some geometric properties of operator spaces

#### Abstract

In this article, we study some geometric properties like parallelism, orthogonality, and semirotundity 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 semirotund points (semirotund spaces) which generalizes the notion of exposed points (strictly convex spaces). We further study semirotund operators and prove that $\mathbb{B}(\mathbb{X},\mathbb{Y})$ is a semirotund space which is not strictly convex if $\mathbb{X},\mathbb{Y}$ are finite-dimensional Banach spaces and $\mathbb{Y}$ is strictly convex.

#### Article information

Source
Banach J. Math. Anal., Volume 13, Number 1 (2019), 174-191.

Dates
Accepted: 22 June 2018
First available in Project Euclid: 4 December 2018

https://projecteuclid.org/euclid.bjma/1543914019

Digital Object Identifier
doi:10.1215/17358787-2018-0021

Mathematical Reviews number (MathSciNet)
MR3892339

Zentralblatt MATH identifier
07002037

#### Citation

Mal, Arpita; Sain, Debmalya; Paul, Kallol. On some geometric properties of operator spaces. Banach J. Math. Anal. 13 (2019), no. 1, 174--191. doi:10.1215/17358787-2018-0021. https://projecteuclid.org/euclid.bjma/1543914019

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