### GENERALIZATIONS OF THE ALUTHGE TRANSFORM OF OPERATORS

Khalid Shebrawi, MOJTABA BAKHERAD

#### Abstract

Let $A = U |A|$ be the polar decomposition of $A$. TheAluthge 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 fornon-negative continuous functions $f, g$ such that $f(x)g(x)=x\,\,(x\geq0)$. Then, by using 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\vertA\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 and non-decreasing convex function on $[0,\infty )$ and $\tilde{A}_{f,g} =f(|A|) U g(|A|)$. Let A = UjAj be the polar decomposition of A. The Aluthge transformof the operator A, denoted by A~, is dened as A~ = jAj12UjAj12 . In this paper, rst wegeneralize the denition of Aluthge transform for non-negative continuous functionsf; g such that f(x)g(x) = x (x 0). Then, by using this denition, we get somenumerical radius inequalities. Among other inequalities, it is shown that if A isbounded linear operator on a complex Hilbert space H , thenh (w(A)) 14

h????g2 (jAj)+ h????f2 (jAj)

+12hwA~f;g;where f; g are non-negative continuous functions such that f(x)g(x) = x (x 0), h isa non-negative and non-decreasing convex function on [0;1) and A~f;g = f(jAj)Ug(jAj).

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