Commuting maps with the Mean Transform
F. Chabbabi, M. Mbekhta · Contemporary mathematics - American Mathematical Society · 2019
Given a bounded operator T ∈ B ( H ) T \in \mathcal {B}(H) , H H a (complex) Hilbert space, let T = V | T | T= V\vert T\vert be the polar decomposition of T T . The Mean transform of the operator T T is defined by M ( T ) := 1 2 ( V | T | + | T | V ) . \mathcal {M}(T):=\frac {1}{2}(V|T|+|T|V). In the present paper, we give a complete characterization of the bijective maps Φ : B ( H ) → B ( K ) \Phi :\ B(H) \to \mathcal {B}(K) , where H , K H,K are Hilbert spaces, that commutes with the mean transform under product. More precisely we show that : M ( Φ ( A ) Φ