## Banach Journal of Mathematical Analysis

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

### On some geometric properties of operator spaces

Arpita Mal, Debmalya Sain, and Kallol Paul

#### 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**

Received: 15 February 2018

Accepted: 22 June 2018

First available in Project Euclid: 4 December 2018

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1215/17358787-2018-0021

**Mathematical Reviews number (MathSciNet)**

MR3892339

**Zentralblatt MATH identifier**

07002037

**Subjects**

Primary: 46B20: Geometry and structure of normed linear spaces

Secondary: 47L05: Linear spaces of operators [See also 46A32 and 46B28]

**Keywords**

norm-parallelism orthogonality semirotund norm attainment

#### 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