Local forms of Bishop-Phelps-Bollob\'{a}s type properties for bilinear maps

TL;DR

This paper explores local Bishop-Phelps-Bollobás properties for bilinear maps, introducing a weaker version and establishing equivalences and duality relations under specific conditions.

Contribution

It defines a weakened weak $ extbf{L}_{o,o}$ property for bilinear maps and proves equivalences with linear functionals, also linking properties of bilinear maps and their adjoints.

Findings

Introduces weak $ extbf{L}_{o,o}$ property for bilinear maps.

Establishes equivalence between properties of bilinear maps and linear functionals.

Shows duality relations between bilinear maps and their adjoint operators.

Abstract

In this paper, we characterize the so called property $\textbf{L}_{o,o}$ as defined by Dantas and Rueda Zoca, for compact, weak-weak continuous bilinear maps. Motivated by this we weaken this property by defining the weak $\textbf{L}_{o,o}$ for bilinear maps. We provide equivalence of the weak $\textbf{L}_{o,o}$ property of $(X\hat{\otimes}_{\pi}Y,\mathbb{R})$ for linear functionals and that of $(X,Y;\mathbb{R})$ for bilinear forms under certain conditions on $X$ and $Y$. Moreover, we have also established that under certain preassigned conditions, if a bilnear map $T$ belongs to a class which satisfies the property $\textbf{L}_{o,o}$ (resp. the weak $\textbf{L}_{o,o}$) for bilinear maps, then $T^*$ is a member of a class of operators which satisfy the property $\textbf{L}_{o,o}$ (resp. the weak $\textbf{L}_{o,o}$) for operators.