Aluthge transforms of operators (Q1587855)

Let \(T\) be a bounded operator in some separable, infinite-dimensional, complex Hilbert space. In terms of its polar decomposition \(T= U|T|\) the Aluthge transform \(\widetilde T\) of \(T\) is given by \(\widetilde T= |T|^{1/2} U|T|^{1/2}\). The article describes connections between \(T\) and \(\widetilde T\). First of all it is proved that the spectra of \(T\) and \(\widetilde T\) coincide. Similar results are shown to hold true also for various other kinds of spectra. Subsequently, the relation between the invariant subspace lattices \(\text{Lat}(T)\) and \(\text{Lat}(\widetilde T)\) of \(T\) and \(\widetilde T\) are studied and it is shown that for quasiaffinities \(\text{Lat}(T)\) is non-trivial if and only if this is ture for \(\text{Lat}(\widetilde T)\).