# A note on \$lambda\$-Aluthge transforms of operators

Document Type: Research Paper

Author

Department of Mathematics and Computer Sciences, Hakim Sabzevari University, P.O. Box 397, Sabzevar, Iran

Abstract

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 emph{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|^{2-2lambda}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|.\$\$

Keywords

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