Perspectives on the ρ-operator radius

doi:10.1016/j.jmaa.2025.130049

Perspectives on the ρ-operator radius

2025-09-01

4

PRE

AI

P

Pintu Bhunia

M

Mohammad Sal Moslehian *

A

Ali Zamani

DOI：10.1016/j.jmaa.2025.130049

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Abstract

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Let rho is an element of (0, 2] and let w(rho)(X) be the rho-operator radius of a Hilbert space operator X. Using techniques involving the Kronecker product, it is shown that 1 + vertical bar 1-rho vertical bar/rho w(X) <= w(rho)(X) <= 2/rho w(X), where w(X) is the numerical radius of X. These bounds for w(rho)(X) are sharper than those presented by J. A. R. Holbrook. Furthermore, the cases of equality are investigated. We prove that the rho-operator radius exposes certain operators as projections. We establish new inequalities for the rho-operator radius, focusing on the sum and product of operators. For the generalized Aluthge transform operator (X) over tilde (t), we prove the inequality: w(rho)(X) <= 1/2 w(rho)((X) over tildet) + 1/rho parallel to X parallel to, for allt is an element of [0, 1]. The derived inequalities extend and generalize several well-known results for the classical operator norm and numerical radius. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Keywords:

rho-Operator radius

Generalized Aluthge transform

Numerical radius

Operator norm

Perspectives on the ρ-operator radius

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