Maps preserving the Aluthge transform of unitarily similar operators

Let B ( H ) 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 E B ( H ) can be uniquely written as T = V T | T L the product of a partial isometry V T E B ( H ) that has the same kernel as that of T and the modulus |T (T*T)12of T . Given a scalar ) E [0, 1], the )-Aluthge transform of any T E B ( H ) is Da(T) :=|T|aVT|T|1-a. In this paper, we obtain the form of all bijective linear maps I' on B ( H ) for which Da (I'(T)) and Da (I'(S)) are unitarily similar whenever T, S E B ( H )are unitarily similar. To achieve this, we characterize all maps I' on B ( H ) for which Da (phi(T)- I'(S)) and Da (T- S ) are unitarily similar for all T, S E B ( H ). Moreover, we obtain the form of all bijective linear maps I' on B ( H ) for which I' (Da(T)) and Da (I'(S)) are unitarily similar whenever T, S E B ( H ) are unitarily similar. Furthermore, a number of related results and consequences is obtained. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.