aluthge transform

Let $widetilde{{C}_{varphi}}$ be the Aluthge transform of composition operator on $L^{2}(Sigma)$. The main result of this paper is characterizations of Aluthge transform of composition operators in some operator classes that are weaker than hyponormal, such as hyponormal, quasihyponormal, paranormal, $*$-paranormal on $L^{2}(Sigma)$. Moreover, to explain the results, we provide several useful related examples to show that $widetilde{{C}_{varphi}}$ lie between these classes.

In this paper, firstly we show that some classical properties for Cauchy dual and Moore-Penrose inverse of composition operators, such as complex symmetric and Aluthge transform on $L^{2}(Sigma)$. Secondly we give a characterization for some operator classes of weak $p$-hyponormal via Moore-Penrose inverse of composition operators. Finally, some examples are then presented to illustrate that, the Moore-Penrose inverse of composition operators lie between these classes.

Let $T$ be a bounded linear operator on a Hilbert space $mathscr{H}$. We say that $T$ has the hyponormal property if there exists a function $f$, continuous on an appropriate set so that $f(|T|)geq f(|T^ast|)$. We investigate the properties of such operators considering certain classes of functions on which our definition is constructed. For such a function $f$ we introduce the $f$-Aluthge transform, $tilde{T}_{f}$. Given two continuous functions $f$ and $g$ with the property  $f(t)g(t)=t$, we also introduce the $(f,g)$-Aluthge transform, $tilde{T}_{(f,g)}$. The features of these transforms are discussed as well.

Let $A=U|A|$ be the polar decomposition of an operator $A$ on a Hilbert space $mathscr{H}$ and $lambdain(0,1)$. The $lambda$-Aluthge transform of $A$ is defined by $tilde{A}_lambda:=|A|^lambda U|A|^{1-lambda}$. In this paper we show that emp {i}) when $mathscr{N}(|A|)=0$, $A$ is self-adjoint if and only if so is $tilde{A}_lambda$ for some $lambdaneq{1over2}$. Also $A$ is self adjoint if and only if $A=tilde{A _lambda^ast$, emph{ii}) if $A$ is normaloid and either $sigma(A)$ has only finitely many distinct nonzero value or $U$ is unitary, then from $A=ctilde{A}_lambda$ for some complex number $c$, we can conclude that $A$ is quasinormal, emph{iii}) if $A^2$ is self-adjoint and any one of the $Re(A)$ or $-Re(A)$ is positive definite then $A$ is self adjoint, emph{iv}) and finally we show that $$opnorm{|A|^{2lambda}|A^ast|^{2- lambda}oplus0}leqopnorm{|A|^{2-2lambda}oplus|A|^{2lambda}} opnorm{tilde{A}_lambdaoplus(tilde{A}_lambda)^ast}$$ where $opnorm{cdot}$ stand for some unitarily invariant norm. From that we conclude that $$||A|^{2lambda |A^ast|^{2-2lambda}|leqmax(|A|^{2lambda},|A|^{2-2lambda})|tilde{A}_lambda|.$$