Two types of hyperinvariant subspaces

Abstract:

Let $A$ be a bounded operator in a Banach space $B$. Suppose that $A$ has the single valued extension property. Given a closed set $F$ in the complexes, define ${\sigma _A}(F)$ to be the set of all $x$ in $B$ such that there is an analytic function $x(\lambda )$ from the complement of $F$ to $B$ with $(A - \lambda I)x(\lambda ) = x$. $A$ is said to have property $Q$ if ${\sigma _A}(F)$ is a closed subset of $B$ for every $F$. Let $A$ be, again, a bounded operator in a Banach space $B$. Given a real number $b$, define ${S_A}(b)$ to be the set of all $x$ in $B$ such that $\exp ( - ct)\exp (At)x$ is a bounded function from the nonnegative reals to $B$ for all $c > b$. $A$ is said to have property $\operatorname {P}$ if ${S_A}(b)$ is a closed subspace of $B$ for all $b$. These two properties are discussed in this paper.