Joint $q$-Numerical Ranges of Operators in Hilbert and Semi-Hilbert Spaces

TL;DR

This paper explores the $q$-numerical range for operator tuples in Hilbert and semi-Hilbert spaces, establishing inequalities and convexity properties to advance understanding of these mathematical structures.

Contribution

It introduces the $q$-numerical range in semi-Hilbert spaces and proves its convexity, extending previous concepts to new space settings.

Findings

Established inequalities for the $q$-numerical radius

Proved convexity of the $q$-numerical range in semi-Hilbert spaces

Extended the concept of $q$-numerical range to semi-Hilbert spaces

Abstract

This paper introduces and investigates the concept of the $q$-numerical range for tuples of bounded linear operators in Hilbert spaces. We establish various inequalities concerning the $q$-numerical radius associated with these operator tuples. Furthermore, we extend our study to define the $q$-numerical range in semi-Hilbert spaces and provide a proof of its convexity. Additionally, we explore several related results in this context.