Two types of hyperinvariant subspaces

Abstract

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