Maps preserving the mean transforms of pairs of unitarily similar operators

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Let B ( H ) be the algebra of all bounded linear operators acting on a complex Hilbert space H . For an operator T ∈ B ( H ) with a polar decomposition T = V T | T | , its mean transform is defined by M ( T ) := 1 2 ( V T | T | + | T | V T ) . In this paper, we obtain the form of all bijective linear maps Φ on B ( H ) for which M ( Φ ( T ) ) and M ( Φ ( S ) ) are unitarily similar whenever T , S ∈ B ( H ) are two unitarily similar operators. To achieve this, we first characterize all surjective maps Φ on B ( H ) satisfying (1) M ( Φ ( T ) − Φ ( S ) ) ∼ u M ( T − S ) , ( T , S ∈ B ( H ) ) . Furthermore, a number of related results and consequences is obtained.