A note on lambda-Aluthge transforms of operators

Let A=U|A| be the polar decomposition of an operator A on a Hilbert space mathscr{H} and lambdain(۰,۱). The lambda-Aluthge transform of A is defined by tilde{A}_lambda:=|A|^lambda U|A|^{۱-lambda}. In this paper we show that emph{i}) when mathscr{N}(|A|)=۰, A is self-adjoint if and only if so is tilde{A}_lambda for some lambdaneq{۱over۲}. 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^۲ 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|^{۲lambda}+|A^ast|^{۲-۲lambda}oplus۰}leqopnorm{|A|^{۲-۲lambda}oplus|A|^{۲lambda}}+ opnorm{tilde{A}_lambdaoplus(tilde{A}_lambda)^ast} where opnorm{cdot} stand for some unitarily invariant norm. From that we conclude that ||A|^{۲lambda}+|A^ast|^{۲-۲lambda}|leqmax(|A|^{۲lambda},|A|^{۲-۲lambda})+|tilde{A}_lambda|.