Reducing decompositions for strictly cyclic operators

Bouldin, Richard

doi:10.1090/S0002-9939-1973-0320777-7

Reducing decompositions for strictly cyclic operators
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Proc. Amer. Math. Soc. 40 (1973), 477-481 Request permission

Abstract:

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

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