Abstract:Given complex numbers $a, b, c$ and a non-negative continuous function $\varphi$ defined on $[0, +\infty)$, consider the $2 \times 2$ matrix $$ M_t = \begin{pmatrix} a & t \\ ct & b\varphi(t) \end{pmatrix}, \quad t \in [0, +\infty). $$ We establish conditions for the strict monotonicity of the norm function $t \mapsto \|M_t\|$. As an application, we characterize the norm attainment of the corresponding block operator matrix $$ T = \begin{pmatrix} aI_H & A \\ cA^* & b\varphi(|A|) \end{pmatrix},$$ where $I_H$ is the identity operator on a Hilbert space $H$ and $A$ is a bounded linear operator from another Hilbert space to $H$.
| Comments: | In the near future, a new section devoted to the numerical range of operators will be added; accordingly, this paper will be expanded |
| Subjects: | Functional Analysis (math.FA) |
| Cite as: | arXiv:2605.25283 [math.FA] |
| (or arXiv:2605.25283v1 [math.FA] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25283 arXiv-issued DOI via DataCite (pending registration) |
From: Qingxiang Xu [view email]
[v1]
Sun, 24 May 2026 22:34:56 UTC (15 KB)
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