Spectral theory for subnormal operators

Abstract:

We give an example of a subnormal operator T such that ${\text {C}} \backslash \sigma (T)$ has an infinite number of components, $\operatorname {int} (\sigma (T))$ has two components U and V, and T cannot be decomposed with respect to U and V. That is, it is impossible to write $T = {T_1} \oplus {T_2}$ with $\sigma ({T_1}) = \overline U$ and $\sigma ({T_2}) = \overline V$. This example shows that Sarason’s decomposition theorem cannot be extended to the infinitely-connected case. We also use Mlak’s generalization of Sarason’s theorem to prove theorems on the existence of reducing subspaces. For example, if X is a spectral set for T and $K \subset X$, conditions are given which imply that T has a nontrivial reducing subspace $\mathcal {M}$ such that $\sigma (T|\mathcal {M}) \subset K$. In particular, we show that if T is a subnormal operator and if $\Gamma$ is a piecewise ${C^2}$ Jordan closed curve which intersects $\sigma (T)$ in a set of measure zero on $\Gamma$, then $T = {T_1} \oplus {T_2}$ with $\sigma ({T_1}) \subset \sigma (T) \cap \overline {\operatorname {ext} (\Gamma )}$ and $\sigma ({T_2}) \subset \sigma (T) \cap \overline {\operatorname {int} (\Gamma )}$.