Quasisimilarity of Hyponormal and Subdecomposable Operators

We prove that quasisimilar subdecomposable operators have equal spectra and quasisimilar subdecomposable operators without eigenvalues have equal essential spectra. Therefore quasisimilar hyponormal operators have equal essential spectra. We also obtain some results on the spectral pictures of cyclic hyponormal operators. An algebra homomorphism π from H∞ (G) to L(H) is a unital representation for T if π(1) = I and π(χ) = T. It is shown that if the boundary of G has zero area measure, then the unital norm continuous representation for a pure hyponormal operator T is unique and is weak star continuous. Consequently, it is proved that every pure hyponormal contraction is in C.0.