Norm optimization problem for linear operators in classical Banach spaces

TL;DR

This paper characterizes when linear operators between classical Banach spaces attain their norm, linking it to the existence of specific maximizing sequences, and explores the structure of norm-attaining operators.

Contribution

It provides a complete characterization of norm attainment for operators from l_p to l_q spaces and analyzes the lineability of norm-attaining operator sets.

Findings

Operators from l_p to l_q attain their norm iff a non-weakly null maximizing sequence exists.

For p<q, any non-weakly null maximizing sequence has a norm-convergent subsequence.

The set of norm-attaining operators exhibits lineability properties.

Abstract

The main result of the paper shows that, for 1<p and 1<=q, a linear operator T from l_p to l_q attains its norm if, and only if, there exists a not weakly null maximizing sequence for T (counterexamples can be easily constructed when p=1). For 1<p (and q different from p), as a consequence of the previous result we show that any not weakly null maximizing sequence for a norm attaining operator T from l_p to l_q has a norm-convergent subsequence (and this result is sharp in the sense that it is not valid if p=q). We also investigate lineability of the sets of norm-attaining and non-norm attaining operators.