On invariant subspaces of operators in the class

In this paper, we consider invariant subspaces of operators in the class θ , which is the set of operators T such that T ∗ T and T + T ∗ commute. It is shown that every operator in the class θ 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 θ which are reductive, i.e., their invariant subspaces are reducing. In addition, we give a condition on spectra of operators in the class θ which gives some information about invariant subspaces.