On the geometry of $G$-norm
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866908905306587136 |
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| author | Dey, Lakshmi Kanta Pal, Subhadip |
| author_facet | Dey, Lakshmi Kanta Pal, Subhadip |
| contents | Let $X$ and $Y$ be Banach spaces and let $G \in L(X,Y)$ with $\|G\|=1$. We study the geometry of $G$-(semi-)norm on $L(X,Y)$, defined by
\[
\|T\|_G := \inf_{δ>0}\sup\{\|Tx\|: \|x\|=1, \|Gx\|>1-δ\},
\]
considering it as a norm ($G$-norm), and further explore the associated numerical indices. In particular, we characterize relative spear operators, that is, operators for which the numerical radius with respect to $G$ coincides with the $G$-norm. Relations among the numerical indices and their invariance under isometric isomorphisms are established. We further obtain a description of the dual unit ball of $(L(X,Y),\|\cdot\|_G)$ and characterize smooth points of its unit ball. In finite-dimensional Hilbert spaces, we prove that relative spear operators are partial isometries. Finally, we establish some equivalent criteria for which the $G$-norm is achieved by the norm attainment set of a norm-attaining operator $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_20796 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the geometry of $G$-norm Dey, Lakshmi Kanta Pal, Subhadip Functional Analysis {Primary 46B20, Secondary 46B04, 47B01 Let $X$ and $Y$ be Banach spaces and let $G \in L(X,Y)$ with $\|G\|=1$. We study the geometry of $G$-(semi-)norm on $L(X,Y)$, defined by \[ \|T\|_G := \inf_{δ>0}\sup\{\|Tx\|: \|x\|=1, \|Gx\|>1-δ\}, \] considering it as a norm ($G$-norm), and further explore the associated numerical indices. In particular, we characterize relative spear operators, that is, operators for which the numerical radius with respect to $G$ coincides with the $G$-norm. Relations among the numerical indices and their invariance under isometric isomorphisms are established. We further obtain a description of the dual unit ball of $(L(X,Y),\|\cdot\|_G)$ and characterize smooth points of its unit ball. In finite-dimensional Hilbert spaces, we prove that relative spear operators are partial isometries. Finally, we establish some equivalent criteria for which the $G$-norm is achieved by the norm attainment set of a norm-attaining operator $G$. |
| title | On the geometry of $G$-norm |
| topic | Functional Analysis {Primary 46B20, Secondary 46B04, 47B01 |
| url | https://arxiv.org/abs/2603.20796 |