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arXiv · 2302.00550

A note on the generalized maximal numerical range of operators

Abderrahim Baghdad·El Hassan Benabdi·Kais Feki
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Abstract

The paper considers some new properties of the so-called $A$-maximal numerical range of operators, denoted by $W_{\max}^A(\cdot)$, where $A$ is a positive bounded linear operator acting on a complex Hilbert space $\mathcal{H}$. Some characterizations of $A$-normaloid operators are also given. In particular, we extend a recent recent by Spitkovsky in [Oper. Matrices, 13, 3(2019)]. Namely, it is shown that an $A$-bounded linear operator $T$ acting on $\mathcal{H}$ is $A$-normaloid if and only if $W_{\max}^A(T)\cap \partial W_A(T)\neq\varnothing$. Here $\partial W_A(T)$ stands for the boundary of $A$-numerical range of $T$. Some new $A$-numerical radius inequalities generalizing and improving earlier well-known results are also given.

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Abderrahim Baghdad, El Hassan Benabdi, Kais Feki. 2023-01-31. A note on the generalized maximal numerical range of operators. https://arxiv.org/abs/2302.00550

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