Aluthge Transforms and Common Invariant Subspaces for a Commuting $\varvec{n}$-Tuple of Operators

Abstract

It is well known that a bounded operator with dense range has a nontrivial invariant subspace if and only if its Aluthge transform does. Recently, R. Curto and Jasang Yoon have introduced the toral and spherical Aluthge transforms for commuting pairs (with particular emphasis on spherically quasinormal and spherically isometric 2-variable weighted shifts) and studied their basic properties. In this paper, we first investigate nontrivial common invariant subspaces between the toral (resp. spherical) Aluthge transform and the original n-tuple of bounded operators with dense ranges. We next study the sets of common invariant subspaces among them. Finally, we give an example which shows that the sets of common invariant subspaces between two Aluthge transforms (toral and spherical) are not isomorphic, even though they are equivalent with respect to the property of existence of a common nontrivial invariant subspace.