Numerical radius inequalities involving commutators of $G_{1}$ operators

TL;DR

This paper establishes new inequalities for the numerical radius involving commutators of $G_{1}$ operators and analytic functions, expanding the theoretical understanding of operator bounds in Hilbert spaces.

Contribution

It introduces novel numerical radius inequalities for commutators of $G_{1}$ operators with analytic functions, providing explicit bounds and conditions.

Findings

Derived inequality for $w(f(A)X+Xar{f}(A))$ involving $w(X-AXA^{*})$

Established bounds for $G_{1}$ operators with spectrum in the unit disk

Extended numerical radius inequalities to classes of analytic functions

Abstract

We prove numerical radius inequalities involving commutators of $G_{1}$ operators and certain analytic functions. Among other inequalities, it is shown that if $A$ and $X$ are bounded linear operators on a complex Hilbert space, then \begin{equation*} w(f(A)X+X\bar{f}(A))\leq {\frac{2}{d_{A}^{2}}}w(X-AXA^{\ast }), \end{equation*} where $A$ is a $G_{1}$ operator with $\sigma (A)\subset \mathbb{D}$ and $f$ is analytic on the unit disk $\mathbb{D}$ such that $\textrm{{Re}}(f)>0$ and $f(0)=1$.