Aluthge transforms of operators

Abstract

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