The Bishop-Phelps-Bollob\'as properties in complex Hilbert spaces

TL;DR

This paper investigates a strengthened version of the Bishop-Phelps-Bollobás property for various classes of operators on complex Hilbert spaces, focusing on norm attainment and numerical radius, with new results for multiple operator classes.

Contribution

It introduces and studies the Bishop-Phelps-Bollobás point property for different operator classes on complex Hilbert spaces, extending the understanding of norm attainment and numerical radius.

Findings

Established the property for self-adjoint, unitary, and normal operators.

Extended results to compact and Schatten-von Neumann operators.

Solved analogous problems using the numerical radius.

Abstract

In this paper we consider a stronger property than the Bishop-Phelps-Bollob\'{a}s property for various classes of operators on a complex Hilbert space. The Bishop-Phelps-Bollob\'as {\it point} property for some class $\mathcal{A} \subset \mathcal{L}(H)$ says that if one starts with a norm one operator $T$ belonging to $\mathcal{A}$, which almost attains its norm at some norm one vector $x_0$, then there is a new operator $S$, belonging to the same class $\mathcal{A}$, which is close to $T$ and attains its norm at the same vector $x_0$. We study it for classical operators on a complex Hilbert spaces such as self-adjoint, anti-symmetric, unitary, compact, normal, and Schatten-von Neumann operators. We also solve analogous problems by replacing the norm of an operator by its numerical radius.