On the convexity of spatial numetical range in normed algebras

TL;DR

This paper investigates whether the spatial numerical range in normed algebras is always convex, confirming convexity in unital cases and providing evidence for non-unital cases through specific examples.

Contribution

The authors prove convexity of the spatial numerical range in several non-unital Banach algebras, extending known results beyond unital cases.

Findings

Convexity holds in unital normed algebras.

Convexity may fail in some non-unital algebras, but is confirmed in several cases.

The problem remains open for general non-unital normed algebras.

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.