The spectrum of a superstable operator and coanalytic families of operators

Abstract

We first show that for an infinite dimensional Banach space $X$, the unitary spectrum of any superstable operator is countable. In connection with descriptive set theory, we show that if $X$ is separable, then the set of stable operators and the set of power bounded operators are Borel subsets of $L(X)$ (equipped with the strong operator topology), while the set $\mathcal{S}'(X)$ of superstable operators is coanalytic. However, $\mathcal{S}'(X)$ is a Borel set if $X$ is a superreflexive and hereditarily indecomposable space. On the other hand, if $X$ is superreflexive and $X$ has a complemented subspace with unconditional basis or, more generally, if $X$ has a polynomially bounded and not superstable operator, then the set $\mathcal{S}'(X)$ is non Borel.