Smooth points in operator spaces and some Bishop-Phelps-Bollob$\acute{a}$s type theorems in Banach spaces

TL;DR

This paper characterizes smooth points in operator spaces and establishes new Bishop-Phelps-Bollobás type theorems in Banach spaces, focusing on reflexive Kadets-Klee spaces and compact operators.

Contribution

It introduces approximate norm attainment sets, characterizes strong BPB properties, and explores uniform BPB approximations, extending previous results in the field.

Findings

Complete characterization of smooth points in compact operator spaces.

Conditions under which pairs of Banach spaces have sBPBp for compact operators.

Connections established between smooth points, BPB properties, and operator space geometry.

Abstract

We introduce the notion of approximate norm attainment set of a bounded linear operator between Banach spaces and use it to obtain a complete characterization of smooth points in the space of compact linear operators, provided the domain space is reflexive and Kadets-Klee. We also apply the concept to characterize strong BPB property (sBPBp) of a pair of Banach spaces. We further introduce uniform $ \epsilon- $BPB approximation of a bounded linear operator and uniform strong BPB property (uniform sBPBp) with respect to a given family of norm one linear operators and explore some of the relevant properties to illustrate its connection with earlier studies on Bishop-Phelps-Bollob$\acute{a}$s type theorems in Banach spaces. It is evident that our study has deep connections with the study of smooth points in operator spaces. We obtain a complete characterization of uniform sBPBp for a pair of Banach spaces, with respect to a given family of norm one bounded linear operators between them. As the final result of this paper, we prove that if $ \mathbb{X} $ is a reflexive Kadets-Klee Banach space and $ \mathbb{Y} $ is any Banach space, then the pair $ (\mathbb{X},\mathbb{Y}) $ has sBPBp for compact operators. Our results extend, complement and improve some of the earlier results in this context.