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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2605.01397 |
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| _version_ | 1866914526256955392 |
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| author | Raposo Jr., Anselmo Ribeiro, Geivison |
| author_facet | Raposo Jr., Anselmo Ribeiro, Geivison |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_01397 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Weak Minimizing Property and the Compact Perturbation Property for the Minimum Modulus Raposo Jr., Anselmo Ribeiro, Geivison Functional Analysis 46B20 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. |
| title | Weak Minimizing Property and the Compact Perturbation Property for the Minimum Modulus |
| topic | Functional Analysis 46B20 |
| url | https://arxiv.org/abs/2605.01397 |