Further bounds on $p$-numerical radii of operators via generalized Aluthge transform

TL;DR

This work addresses bounding the $p$-numerical radius $w_p(T)$ of operators in Schatten classes by leveraging the generalized $(f,g)$-Aluthge transform $\widetilde{T}_{f,g}$. It derives a central inequality $w_p(T)\le 2^{\frac{1}{p}-1}w_p(\widetilde{T}_{f,g})+2^{\frac{1}{p}-2}\|f^2(|T|)+g^2(|T|)\|_p$, and provides numerous corollaries, including the standard choice $f(x)=x^{1-t}$, $g(x)=x^{t}$, with $t\in[0,1]$, plus $p=\infty$ refinements. The Main Results extend to operator matrices and sums via the polar decomposition and the $0TS0$ structure, yielding Heinz-type and Kittaneh-type refinements for $w_p$ and related norms. Overall, the paper unifies and strengthens existing Yamazaki-type inequalities and related bounds, offering sharper control of $p$-numerical radii in the Hilbert space setting.

Abstract

The main aim of this article is to establish several $p$-numerical radius inequalities via the $(f,g)$-Aluthge transform of Hilbert space operators and operator matrices. Furthermore, various classical numerical radius and norm inequalities for Hilbert space operators are also discussed. The bounds obtained in this work improve upon several well-known earlier results.

Theorems & Definitions (28)

• Lemma 1.1
• Lemma 1.2
• Lemma 1.3
• Lemma 1.4
• Lemma 1.5
• Theorem 2.1
• proof
• Corollary 2.1
• Remark 2.1
• Remark 2.2