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arXiv · 2306.16141

On the convexity of spatial numetical range in normed algebras

H. V. Dedania·A. B. Patel
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Abstract

In this article, we address the following question: Is it true that the spatial numerical range (SNR) $V_A(a)$ of an element $a$ in a normed algebra $(A, \|\cdot\|)$ is always convex? If the normed algebra is unital, then it is convex \cite[Theorem 3, P.16]{BoDu:71}. In non-unital case, we believe that the problem is still open and its answer seems to be negative. In search of such a normed algebra, we have proved that the SNR $V_A(a)$ is convex in several non-unital Banach algebras.

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BibTeXRIS

H. V. Dedania, A. B. Patel. 2023-06-28. On the convexity of spatial numetical range in normed algebras. https://arxiv.org/abs/2306.16141

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