Spectral theory for subnormal operators

Abstract

We give an example of a subnormal operator T such that C ∖ σ ( T ) {\text {C}}\,\backslash \,\sigma (T) has an infinite number of components, int ⁡ ( σ ( T ) ) \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 ⊕ T 2 T\, = \,{T_1}\, \oplus \,{T_2} with σ ( T 1 ) = U ¯ \sigma ({T_1})\, = \,\overline U and σ ( T 2 ) = V ¯ \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 ⊂ X K\, \subset \,X , conditions are given which imply that T has a nontrivial reducing subspace M \mathcal {M} such that σ ( T | M ) ⊂ K \sigma (T|\mathcal {M})\, \subset \,K . In particular, we show that if T is a subnormal operator and if Γ \Gamma is a piecewise C 2 {C^2} Jordan closed curve which intersects σ ( T ) \sigma (T) in a set of measure zero on Γ \Gamma , then T = T 1 ⊕ T 2 T\, = \,{T_1}\, \oplus \,{T_2} with σ ( T 1 ) ⊂ σ ( T ) ∩ ext ⁡ ( Γ ) ¯ \sigma ({T_1})\, \subset \,\sigma (T)\, \cap \,\overline {\operatorname {ext} (\Gamma )} and σ ( T 2 ) ⊂ σ ( T ) ∩ int ⁡ ( Γ ) ¯ \sigma ({T_2})\, \subset \,\sigma (T)\, \cap \,\overline {\operatorname {int} (\Gamma )} .