Abstract: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 $\epsilon$-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 $\epsilon$-Birkhoff orthogonal to a subspace of $\mathcal{B(H,K)}$, when $A$ is compact.
| Subjects: | Functional Analysis (math.FA) |
| MSC classes: | 15A60, 47L10, 46G05, 46B20, 47A30 |
| Cite as: | arXiv:2505.06925 [math.FA] |
| (or arXiv:2505.06925v4 [math.FA] for this version) | |
| https://doi.org/10.48550/arXiv.2505.06925 arXiv-issued DOI via DataCite |
From: Susmita Seal [view email]
[v1]
Sun, 11 May 2025 10:00:44 UTC (18 KB)
[v2]
Wed, 16 Jul 2025 09:42:30 UTC (17 KB)
[v3]
Thu, 27 Nov 2025 16:19:06 UTC (19 KB)
[v4]
Fri, 22 May 2026 07:41:32 UTC (14 KB)
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