Almost isolated spectral parts and invariant subspaces

Abstract:

Let T be an operator with spectrum $\sigma (T)$ on a Hilbert space. A compact subset E of $\sigma (T)$ is called a disconnected part of $\sigma (T)$ if, for some bounded open set A, E is the closure of $\sigma (T) \cap A$ and $\sigma (T) - E$ is the union of the isolated parts of $\sigma (T)$ lying completely outside the closure of A. The set E is called an almost isolated part of $\sigma (T)$ if, in addition, the set A can be chosen so as to have a rectifiable boundary $\partial A$ on which the subset $\sigma (T) \cap \partial A$ has arc length measure 0. The following results are obtained. If T is subnormal and if E is a disconnected part of $\sigma (T)$ then there exists a reducing subspace $\mathfrak {M}$ of T for which $\sigma (T|\mathfrak {M}) = E$. If ${T^\ast }$ is hyponormal and if E is an almost isolated part of $\sigma (T)$ then there exists an invariant subspace $\mathfrak {M}$ of T for which $\sigma (T|\mathfrak {M}) = E$. An example is given showing that if T is arbitrary then an almost isolated part of $\sigma (T)$ need not be the spectrum of the restriction of T to any invariant subspace.