Abstract. Let $T$ and $R$ be absolutely continuous polynomially bounded operators, that is, $H^\infty $-calculus is well-defined for them, and let $X$ and $Y$ be quasiaffinities which intertwine $T$ and $R$: $XT=RX$, $YR=TY$. If there exists a function $g\in H^\infty $ such that $XY=g(R)$, then $\sigma(T)=\sigma(R)$ and $\sigma_{\text{e}}(T)=\sigma_{\text{e}}(R)$. Also, a generalization of the result for contractions of K. Takahashi [14] is given: if a polynomially bounded operator $T$ is a quasiaffine transform of a unilateral shift $S$ of finite multiplicity, then $\sigma_{\text{e}}(T)=\sigma_{\text{e}}(S)$ and $\mathop{\rm ind}T=\mathop{\rm ind}S$, where $\text{ind}$ is the Fredholm index.

DOI: 10.14232/actasm-013-064-8

AMS Subject Classification (1991): 47A10, 47A60, 47A99

Keyword(s): polynomially bounded operator, quasisimilarity, quasiaffine transform, spectrum, essential spectrum, unilateral shift

Received September 16, 2013, and in final form January 24, 2014. (Registered under 64/2013.)