On the spectra of the restrictions of an operator

Abstract:

Let T be a bounded linear operator from a complex Banach space $\mathfrak {X}$ into itself and let $\mathfrak {M}$ be a closed invariant subspace of T. Let $T|\mathfrak {M}$ denote the restriction of T to $\mathfrak {M}$ and let $\sigma$ denote the spectrum of an operator. The main results say that: (1) If $\mathfrak {X}$ is the closed linear span of a family $\{ {\mathfrak {M}_v}\}$ of invariant subspaces, then every component of $\sigma (T)$ intersects the closure of the set ${ \cup _v}\sigma (T|{\mathfrak {M}_v})$ and every point of $\sigma (T)\backslash { \cup _v}\sigma (T|{\mathfrak {M}_v})$ is an approximate eigenvalue of T. (2) If $\mathfrak {X}$ is the closed linear span of a finite family $\{ {\mathfrak {M}_1}, \ldots ,{\mathfrak {M}_n}\}$ of invariant subspaces, and the spectra $\sigma (T|{\mathfrak {M}_j}),j = 1,2, \ldots ,n$, are pairwise disjoint, then $\mathfrak {X}$ is actually equal to the algebraic direct sum of the ${\mathfrak {M}_j}$’s, the ${\mathfrak {M}_j}$’s are hyperinvariant subspaces of T and $\sigma (T) = \cup _{j = 1}^n\sigma (T|{\mathfrak {M}_j})$. This last result is sharp in a certain specified sense. The results of (1) have a “dual version” $(1’)$; (1) and $(1’)$ are applied to analyze the spectrum of an operator having a chain of invariant subspaces which is “piecewise well-ordered by inclusion", extending in several ways recent results of J. D. Stafney on the spectra of lower triangular matrices.