On invariant subspaces of operators in the class θθ

In this paper, we consider invariant subspaces of operators in the class θθ, which is the set of operators TT such that T∗TT∗T and T+T∗T+T∗ commute. It is shown that every operator in the class θθ such that the outer boundary of its spectrum is the outer boundary of a Carathéodory domain has a nontrivial invariant subspace. We also give a family of operators in the class θθ which are reductive, i.e., their invariant subspaces are reducing. In addition, we give a condition on spectra of operators in the class θθ which gives some information about invariant subspaces.