Abstract
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.
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