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 onl...

In this paper, we prove that each of the following functions is convex on R: f(t) = wN(AtXA1?t ? A1?tXAt), g(t) wN(AtXA1?t), and h(t) wN(AtXAt) where A > 0, X Mn N(.) a unitarily invariant norm onMn. Consequently, answer positively question concerning convexity function t w(AtXAt) proposed by in (2018). We provide some generalizations extensions wN(.) using Kwong functions. More precisely, w...