Maps preserving the Aluthge transform of unitarily similar operators

Abstract

Let $B\left(H\right)$ be the algebra of all bounded linear operators acting on an infinite-dimensional separable complex Hilbert space $H$. The polar decomposition theorem asserts that every operator $T\in B\left(H\right)$ can be uniquely written as $T=V_T\left|T\right|$, the product of a partial isometry $V_T\in B\left(H\right)$ that has the same kernel as that of T and the modulus $\left|T\right|:={\left(T^{⁎}T\right)}^{\frac{1}{2}}$ of T. Given a scalar $\lambda \in [0,1]$, the λ-Aluthge transform of any $T\in B\left(H\right)$ is ${\Delta }_{\lambda }\left(T\right):=\left|T{|}^{\lambda }{V}_T\right|T{|}^{1-\lambda }$. In this paper, we obtain the form of all bijective linear maps Φ on $B\left(H\right)$ for which ${\Delta }_{\lambda }\left(\Phi \right(T\left)\right)$ and ${\Delta }_{\lambda }\left(\Phi \right(S\left)\right)$ are unitarily similar whenever $T,S\in B\left(H\right)$ are unitarily similar. To achieve this, we characterize all maps Φ on $B\left(H\right)$ for which ${\Delta }_{\lambda }\left(\varphi \right(T)-\Phi (S\left)\right)$ and ${\Delta }_{\lambda }(T-S)$ are unitarily similar for all $T,S\in B\left(H\right)$. Moreover, we obtain the form of all bijective linear maps Φ on $B\left(H\right)$ for which $\Phi \left({\Delta }_{\lambda }\right(T\left)\right)$ and ${\Delta }_{\lambda }\left(\Phi \right(S\left)\right)$ are unitarily similar whenever $T,S\in B\left(H\right)$ are unitarily similar. Furthermore, a number of related results and consequences is obtained.