Subnormal operators, Toeplitz operators and spectral inclusion

Abstract:

Let $S$ be a subnormal operator on the Hilbert space $H$, and let $N = \int z \;dE(z)$ be its minimal normal extension on $K$. Let $\mu$ be a scalar spectral measure for $N$. If $f \in {L^\infty }(\mu )$, define ${T_f} = Pf(N){|_H}:\;H \to H$, where $P:K \to H$ denotes orthogonal projection. $S$ has the ${C^ \ast }$-Spectral Inclusion Property (${C^ \ast }$-SIP) if $\sigma (f(N)) \subseteq \sigma ({T_f})$, for all $f \in C(\sigma (N))$, and $S$ has the ${W^\ast }$-Spectral Inclusion Property (${W^\ast }$-SIP) if $\sigma (f(N)) \subseteq \sigma ({T_f})$, for all $f \in {L^\infty }(\mu )$. It is shown that $S$ has the ${C^\ast }$-SIP if and only if $\sigma (N) = \Pi (S)$, the approximate point spectrum of $S$. This is equivalent to requiring that $E(\Delta )K$ have angle $0$ with $H$, for all nonempty, relatively open $\Delta \subseteq \sigma (N)$. $S$ has the ${W^\ast }$-SIP if this angle condition holds for all proper Borel subsets of $\sigma (N)$. If $S$ is pure and has the ${C^\ast }$ or ${W^\ast }$-SIP, then it is shown that $\sigma (f(N)) \subseteq {\sigma _e}({T_f})$, for all appropriate $f$.