Weak Minimizing Property and the Compact Perturbation Property for the Minimum Modulus

TL;DR

The paper investigates properties related to the minimum modulus of operators, proving that certain pairs of Banach spaces do not satisfy the Compact Perturbation Property, with implications for operator theory.

Contribution

It provides a negative answer to whether the pair (c0,c0) has the CPPm, demonstrating a non-min-attaining operator can be increased by a compact perturbation.

Findings

(c0,c0) does not have the CPPm.

Constructive proof of a non-min-attaining operator with increased minimum modulus.

Failure of CPPm extends to pairs (X,X) with specific non-reflexive structures.

Abstract

For an operator $T:X\to Y$, denote $m(T)=\inf\{\|Tx\|:x\in S_X\}$. A sequence $(x_n)$ in $S_X$ is said to be minimizing for $T$ if $\|Tx_n\|\to m(T)$. The weak minimizing property (WmP), introduced by Chakraborty, requires that every operator admitting a non-weakly null minimizing sequence attains its minimum modulus. More recently, Han~\cite{Han2026} introduced the Compact Perturbation Property for the minimum modulus (CPPm), which requires that for every operator $T:X\to Y$ that does not attain its minimum modulus, \[ \sup_{K\in\mathcal{K}(X,Y)} m(T+K)=m(T). \] In~\cite{Han2026}, it is shown that $(\ell_1,\ell_1)$ fails both properties, while $(c_0,c_0)$ fails the WmP. However, whether $(c_0,c_0)$ has the CPPm was left open (Problem~3.6). In this paper, we give a negative answer to this question by proving that $(c_0,c_0)$ does not have the CPPm. The proof is constructive, exhibiting a non-min-attaining operator whose minimum modulus is strictly increased by a rank-one compact perturbation. Moreover, we show that this phenomenon is not specific to $c_0$: if $X=\mathbb{K}\oplus_\infty Y$ with $Y$ non-reflexive, then the pair $(X,X)$ fails the CPPm.