Polar decomposition, Aluthge and mean transforms
Fadil Chabbabi, Mostafa Mbekhta · Contemporary mathematics - American Mathematical Society · 2020
In this paper, we give a new proof of the existence and uniqueness of the polar decomposition T = V | T | T = V\vert T\vert of a bounded linear operator T T on a complex Hilbert space H H . We show that the polar part of the polar decomposition of T T is given by an explicit formula: V = ∫ 0 ∞ T exp ( − s T ∗ T ) | T | d s . \begin{equation*} V = \int _0^{\infty }T\exp \left (-sT^*T\right ) \vert T\vert ds. \end{equation*} On the other hand, we establish new results on the Aluthge and mean transforms of a bounded linear operator T T acting on H H . Among other things, we show under some conditions that Aluthge transform of T T has closed range if and only if T T itself has closed range. We also prove that the mean transform preserves the class of compact operators and Schatten p p -ideal. We end this paper by asking several open questions.