Inequalities for the generalized numerical radius

TL;DR

This paper develops new inequalities for the generalized numerical radius of operator matrices, extending convexity and log-convexity results, and applies Schatten p-norm inequalities to derive bounds, advancing the theoretical understanding of operator norms.

Contribution

It introduces generalized inequalities for the numerical radius, extends convexity properties, and applies Schatten p-norm inequalities to operator matrices, broadening the scope of existing results.

Findings

Established new inequalities for the generalized numerical radius.

Extended convexity and log-convexity results to the generalized case.

Provided Schatten p-norm inequalities for operator matrices.

Abstract

In the present paper, we provide several inequalities for the generalized numerical radius of operator matrices as introduced by A. Abu-omar and F. Kittaneh in [3]. We generalize the convexity and the log-convexity results obtained by M. Sababheh in [12] for the case of the numerical radius to the case of the generalized numerical radius. We illustrate our work by providing a positive answer for the question addressed in [12] for the convexity of a certain matrix operator function. Moreover, and motivated by the results of A. Aldalabih and F. Kittaneh in [2] for the case of Hilbert-Schmidt numerical radius norm, we use some Schatten $p$-norm inequalities for partitioned $2\times 2$ block-matrices to provide several Schatten $p$-norm numerical radius inequalities.