Aluthge transforms of operators

Abstract

Associated with every operator T on Hilbert space is its Aluthge transform T̃ (defined below). In this note we study various connections between T and T̃, including relations between various spectra, numerical ranges, and lattices of invariant subspaces. In particular, we show that if T̃ has a nontrivial invariant subspace, then so does T, and we give various applications of our results.