a note on λ-aluthge transforms of operators

Let a=u|a| be the polar decomposition of an operator a on a hilbert space mathscr{h} and λϵ(0,1) . the λaluthge transform of a is defined by ã_λ:=|a|^λ u|a|^1-λ . in this paper we show that emph i) when mathscr{n}(|a|)=0, a is selfadjoint if and only if so is ãλ for some λ≠1/2. also a is self adjoint if and only if a=ã_λ ^*,ii) if a is normaloid and either σ(a) has only finitely many distinct nonzero value or u is unitary, then from a=cã_λ for some complex number c, we can conclude that a is quasinormal, iii) if a^2 is selfadjoint and any one of the re(a) or re(a) is positive definite then a is self-adjoint, iv) and finally we show that ||| |a|^2λ+|a^*|^2-2λ oplus0||| |a|^2-2λoplus |a|^2λ||| + |||ã_λ oplus (ã_λ)*|||where ||| . ||| stand for some unitarily invariant norm. from that we conclude that ||| |a|^2λ+|a^*|^2-2λ || ≤ max (|| |a|^2λ ,|| |a|2-^2λ)+||ã_λ||.