Numerical radius inequalities of operator matrices from a new norm on $\mathcal{B}(\mathcal{H})$

TL;DR

This paper introduces bounds for the $(ta,eta)$-norm of operator matrices and uses these to derive generalized estimates for their numerical radius and operator norm.

Contribution

It develops new upper bounds for the $(ta,eta)$-norm of operator matrices and applies these to improve bounds on numerical radius and operator norm.

Findings

Derived upper bounds for the $(ta,eta)$-norm of operator matrices.

Established generalized bounds for numerical radius of operator matrices.

Provided estimates that extend existing bounds for operator norms.

Abstract

This paper is a continuation of a recent work on a new norm, christened the $ (\alpha, \beta)$-norm, on the space of bounded linear operators on a Hilbert space. We obtain some upper bounds for the said norm of $n\times n$ operator matrices. As an application of the present study, we estimate bounds for the numerical radius and the usual operator norm of $n\times n$ operator matrices, which generalize the existing ones.