Weak minimizing property and reflexivity

TL;DR

This paper investigates the weak minimizing property (WmP) in Banach space pairs, establishing conditions under which reflexivity implies WmP and exploring the impact of subspace structures on this property.

Contribution

It provides new characterizations of WmP for pairs of infinite-dimensional separable Banach spaces, linking reflexivity and subspace isomorphisms to the property.

Findings

If (X,Y) has WmP, then X is reflexive.

If X is reflexive and Y does not contain an isomorphic copy of X, then (X,Y) has WmP.

Existence of an equivalent norm on Y can prevent WmP when Y contains an isomorphic copy of X.

Abstract

For an operator T from X to Y denote m(T) the infimum of $||Tx||$ on the unit sphere $S_X$ of X. A sequence $(x_n)$ in $S_X$ is said to be minimizing for T if $||Tx_n||$ tends to m(T). In 2020 U. S. Chakraborty introduced and studied the following weak minimizing property (WmP): a pair (X,Y) of Banach spaces is said to have the WmP if, for every bounded linear operator $T: X \to Y$ that admits a non-weakly null minimizing sequence, the function $x \mapsto \|Tx\|$ attains its minimum on the unit sphere. We present the following new results about the WmP for pairs of infinite-dimensional separable Banach spaces: (i) If (X,Y) has the WmP, then X is reflexive. (ii) If X is reflexive and Y does not contain isomorphic copies of X, then (X,Y) has the WmP. (iii) If X is reflexive and Y contains an isomorphic copy of X, then there is an equivalent norm on Y such that, for this equivalent norm, (X,Y) does not have the WmP. The first result extends to non-separable X if and only if X possesses a countable total set of functionals.