On Bhatia-Šemrl Property, Strong Subdifferentiability and Essential Norm of Operators on Banach Spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917151416254464 |
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| author | Jayanarayanan, C. R. Rajpopat, Rishit R |
| author_facet | Jayanarayanan, C. R. Rajpopat, Rishit R |
| contents | We investigate the interplay among three key properties of bounded linear operators between Banach spaces: the Bhatia-Šemrl property, strong subdifferentiability and the condition that the essential norm is strictly less than the operator norm. For a Hilbert space $H$ and for $1<p,q<\infty$, we show that for any operator in $B(H)$ and $B(\ell_p, \ell_q)$, the essential norm is strictly less than the operator norm if and only if it is the point of strong subdifferentiability of the norm and its norm-attainment set is compact. Moreover, for operators in these spaces that satisfy the Bhatia-Šemrl property, we show that their essential norm must be strictly less than their operator norm. We also study norm one projections satisfying the Bhatia-Šemrl property and provide examples of operators that possess this property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_15208 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Bhatia-Šemrl Property, Strong Subdifferentiability and Essential Norm of Operators on Banach Spaces Jayanarayanan, C. R. Rajpopat, Rishit R Functional Analysis Primary 46B20, 47B01, Secondary 46G05, 47L05 We investigate the interplay among three key properties of bounded linear operators between Banach spaces: the Bhatia-Šemrl property, strong subdifferentiability and the condition that the essential norm is strictly less than the operator norm. For a Hilbert space $H$ and for $1<p,q<\infty$, we show that for any operator in $B(H)$ and $B(\ell_p, \ell_q)$, the essential norm is strictly less than the operator norm if and only if it is the point of strong subdifferentiability of the norm and its norm-attainment set is compact. Moreover, for operators in these spaces that satisfy the Bhatia-Šemrl property, we show that their essential norm must be strictly less than their operator norm. We also study norm one projections satisfying the Bhatia-Šemrl property and provide examples of operators that possess this property. |
| title | On Bhatia-Šemrl Property, Strong Subdifferentiability and Essential Norm of Operators on Banach Spaces |
| topic | Functional Analysis Primary 46B20, 47B01, Secondary 46G05, 47L05 |
| url | https://arxiv.org/abs/2512.15208 |