Smoothness in the space of bounded linear operators on semi-Hilbert space
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866914014488952832 |
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| author | Barik, Somdatta Ghosh, Souvik Paul, Kallol Sain, Debmalya |
| author_facet | Barik, Somdatta Ghosh, Souvik Paul, Kallol Sain, Debmalya |
| contents | Given a nonzero positive operator $A$ on a Hilbert space $\mathbb{H}$, a semi-inner product is naturally induced on $\mathbb{H}$. In this work, we introduce the notion of \emph{$A$-smoothness} for bounded linear operators on the resulting semi-Hilbert space and investigate its various properties. We provide a comprehensive characterization of the $A$-smoothness for $A$-bounded operators and further analyze the $A$-smoothness of $A$-compact operators in terms of their $A$-norm attainment sets. Utilizing these characterizations, we establish that Gâteaux differentiability of the semi-norm $\|\cdot\|_A$ at an $A$-bounded operator is equivalent to its $A$-smoothness. Furthermore, we characterize the $A$-smoothness of $2\times 2$ block diagonal matrices. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_00335 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Smoothness in the space of bounded linear operators on semi-Hilbert space Barik, Somdatta Ghosh, Souvik Paul, Kallol Sain, Debmalya Functional Analysis 46B20, 46C50, 47L0 Given a nonzero positive operator $A$ on a Hilbert space $\mathbb{H}$, a semi-inner product is naturally induced on $\mathbb{H}$. In this work, we introduce the notion of \emph{$A$-smoothness} for bounded linear operators on the resulting semi-Hilbert space and investigate its various properties. We provide a comprehensive characterization of the $A$-smoothness for $A$-bounded operators and further analyze the $A$-smoothness of $A$-compact operators in terms of their $A$-norm attainment sets. Utilizing these characterizations, we establish that Gâteaux differentiability of the semi-norm $\|\cdot\|_A$ at an $A$-bounded operator is equivalent to its $A$-smoothness. Furthermore, we characterize the $A$-smoothness of $2\times 2$ block diagonal matrices. |
| title | Smoothness in the space of bounded linear operators on semi-Hilbert space |
| topic | Functional Analysis 46B20, 46C50, 47L0 |
| url | https://arxiv.org/abs/2509.00335 |