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a b s t r a c tIn 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.

In this paper it is shown that if an operator T satisfies p(T )p σ (T ) for every polynomial p and the polynomially convex hull of σ (T ) is a Carathéodory region whose accessible boundary points lie in rectifiable Jordan arcs on its boundary, then T has a nontrivial invariant subspace. As a corollary, it is also shown that if T is a hyponormal operator and the outer boundary of σ (T ) has at most finitely many prime ends corresponding to singular points on ∂D and has a tangent at almost every point on each Jordan arc, then T has a nontrivial invariant subspace.

In this note we answer an old question of Brown, Douglas, and Fillmore [L. Brown, R.G. Douglas, P. Fillmore, Unitary equivalence modulo the compact operators and extensions of C * -algebras, in: Proc.

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