Reducing decompositions for strictly cyclic operators

Abstract

If T T is a strictly cyclic operator on H H then H H has a direct sum decomposition H 1 ⊕ H 2 {H_1} \oplus {H_2} where H 1 {H_1} and H 2 {H_2} are invariant under T T if and only if the spectrum of T T is not connected. If λ \lambda is a reducing eigenvalue for the strictly cyclic operator T T then the multiplicity of λ \lambda is one and λ \lambda is an isolated point of the spectrum of T T .