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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.18534 |
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| _version_ | 1866914498753855488 |
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| author | Kadets, Vladimir Ribeiro, Geivison |
| author_facet | Kadets, Vladimir Ribeiro, Geivison |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_18534 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Weak minimizing property and reflexivity Kadets, Vladimir Ribeiro, Geivison Functional Analysis 46B20 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. |
| title | Weak minimizing property and reflexivity |
| topic | Functional Analysis 46B20 |
| url | https://arxiv.org/abs/2604.18534 |