Some generalizations of the Aluthge transform of operators

TL;DR

This paper extends the Aluthge transform to a broader class of functions and derives new numerical radius inequalities for operators on complex Hilbert spaces.

Contribution

It introduces a generalized Aluthge transform using non-negative functions and establishes novel inequalities involving the numerical radius of operators.

Findings

Derived new numerical radius inequalities for generalized Aluthge transforms.

Showed bounds involving convex functions of the numerical radius.

Extended the classical Aluthge transform to a wider functional framework.

Abstract

Let $A = U |A|$ be the polar decomposition of $A$. The Aluthge transform of the operator $A$, denoted by $\tilde{A}$, is defined as $\tilde{A} =|A|^{\frac{1}{2}} U |A|^{\frac{1}{2}}$. In this paper, first we generalize the definition of Aluthge transform for non-negative continuous functions $f, g$ such that $f(x)g(x)=x\,\,(x\geq0)$. Then, by using of this definition, we get some numerical radius inequalities. Among other inequalities, it is shown that if $A$ is bounded linear operator on a complex Hilbert space ${\mathscr H}$, then \begin{equation*} h\left( w(A)\right) \leq \frac{1}{4}\left\Vert h\left( g^{2}\left( \left\vert A\right\vert \right) \right) +h\left( f^{2}\left( \left\vert A\right\vert \right) \right) \right\Vert +\frac{1}{2}h\left( w\left( \tilde{A}_{f,g}\right) \right) , \end{equation*} where $f, g$ are non-negative continuous functions such that $f(x)g(x)=x\,\,(x\geq 0)$, $h$ is a non-negative non-decreasing convex function on $[0,\infty )$ and $\tilde{A}_{f,g} =f(|A|) U g(|A|)$.