Nice surjections on spaces of operators

TL;DR

This paper characterizes nice surjective linear operators on spaces of compact and bounded operators between Banach spaces, establishing conditions under which they are composition operators and exploring their automatic continuity.

Contribution

It provides necessary and sufficient conditions for nice surjections to be composition operators on spaces of operators, extending previous results to Banach spaces and non-canonical forms.

Findings

Characterization of nice surjections as composition operators

Automatic continuity of these operators

Extension of results to ${ m L}^p$ spaces and non-canonical forms

Abstract

A bounded linear operator is said to be nice if its adjoint preserves extreme points of the dual unit ball. Motivated by a description due to Labuschagne and Mascioni \cite{LM} of such maps for the space of compact operators on a Hilbert space, in this article we consider a description of nice surjections on ${\mathcal K}(X,Y)$ for Banach spaces $X,Y$. We give necessary and sufficient conditions when nice surjections are given by composition operators. Our results imply automatic continuity of these maps with respect to other topologies on spaces of operators. We also formulate the corresponding result for ${\mathcal L}(X,Y)$ thereby proving an analogue of the result from \cite{LM} for $L^p$ ($1 <p \neq 2 <\infty$) spaces. We also formulate results when nice operators are not of the canonical form, extending and correcting the results from \cite{KS}.