Inequalities on the spectral radius and the operator norm of Hadamard products of positive operators on sequence spaces

TL;DR

This paper extends and refines existing inequalities related to the spectral radius, operator norm, and numerical radius of Hadamard products of positive operators on sequence spaces, introducing some novel results even for finite matrices.

Contribution

It generalizes previous inequalities for spectral radius and operator norm, and establishes new bounds for the numerical radius of Hadamard products of positive operators.

Findings

Extended inequalities for spectral radius and operator norm.

Proved new bounds for numerical radius.

Some results are novel even for finite non-negative matrices.

Abstract

Relatively recently, K.M.R. Audenaert (2010), R.A. Horn and F. Zhang (2010), Z. Huang (2011), A.R. Schep (2011), A. Peperko (2012), D. Chen and Y. Zhang (2015) have proved inequalities on the spectral radius and the operator norm of Hadamard products and ordinary matrix products of finite and infinite non-negative matrices that define operators on sequence spaces. In the current paper we extend and refine several of these results and also prove some analogues for the numerical radius. Some inequalities seem to be new even in the case of $n\times n$ non-negative matrices.