Subdifferential of the $\mathcal{B(H,K)}$ norm, and approximate orthogonality
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917521115840512 |
|---|---|
| author | Grover, Priyanka Gupta, Krishna Kumar Seal, Susmita |
| author_facet | Grover, Priyanka Gupta, Krishna Kumar Seal, Susmita |
| contents | We present an expression for the right hand derivative of the $\mathcal{B(H,K)}$ norm generalizing the result for $\mathcal{K}=\mathcal{H}$ in [D. J. Ke$\check{\mathrm{c}}$ki$\grave{\mathrm{c}}$, Gateaux derivative of $B(H)$ norm, Proc. Amer. Math. Soc. 133 (2005): 2061--2067]. Using this, we obtain the subdifferential of the $\mathcal{B(H, K)}$ norm. For tuples of operators $\mathbf{A},\mathbf{X}\in$ $\mathcal{B(H, H}^d)$, we give a characterization for $\boldsymbol 0$ to be a best approximation to the subspace $\mathbb C^d \mathbf{X}$, generalizing a similar result for $\mathbb C^d \mathbf{I}$ in [P. Grover, S. Singla, A distance formula for tuples of operators, Linear Algebra Appl. 650 (2022): 267--285]. We define the concept of $ε$-Birkhoff orthogonality to a subspace in a general normed space and derive a characterization in terms of the subdifferential set. Using this, we deduce interesting results for $A\in \mathcal{B(H,K)}$ to be $ε$-Birkhoff orthogonal to a subspace of $\mathcal{B(H,K)}$, when $A$ is compact. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_06925 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Subdifferential of the $\mathcal{B(H,K)}$ norm, and approximate orthogonality Grover, Priyanka Gupta, Krishna Kumar Seal, Susmita Functional Analysis 15A60, 47L10, 46G05, 46B20, 47A30 We present an expression for the right hand derivative of the $\mathcal{B(H,K)}$ norm generalizing the result for $\mathcal{K}=\mathcal{H}$ in [D. J. Ke$\check{\mathrm{c}}$ki$\grave{\mathrm{c}}$, Gateaux derivative of $B(H)$ norm, Proc. Amer. Math. Soc. 133 (2005): 2061--2067]. Using this, we obtain the subdifferential of the $\mathcal{B(H, K)}$ norm. For tuples of operators $\mathbf{A},\mathbf{X}\in$ $\mathcal{B(H, H}^d)$, we give a characterization for $\boldsymbol 0$ to be a best approximation to the subspace $\mathbb C^d \mathbf{X}$, generalizing a similar result for $\mathbb C^d \mathbf{I}$ in [P. Grover, S. Singla, A distance formula for tuples of operators, Linear Algebra Appl. 650 (2022): 267--285]. We define the concept of $ε$-Birkhoff orthogonality to a subspace in a general normed space and derive a characterization in terms of the subdifferential set. Using this, we deduce interesting results for $A\in \mathcal{B(H,K)}$ to be $ε$-Birkhoff orthogonal to a subspace of $\mathcal{B(H,K)}$, when $A$ is compact. |
| title | Subdifferential of the $\mathcal{B(H,K)}$ norm, and approximate orthogonality |
| topic | Functional Analysis 15A60, 47L10, 46G05, 46B20, 47A30 |
| url | https://arxiv.org/abs/2505.06925 |