The spectrum of a superstable operator and coanalytic families of operators

We first show that for an infinite dimensional Banach space X, the unitary spectrum of any superstable operator is countable. In connection with descr iptive set theory, we show that if X is separable, then the set of stable o perators and the set of power bounded operators are Borel subsets of L(X) ( equipped with the strong operator topology), while the set S'(X) of superst able, operators is coanalytic. However, S'(X) is a Borel set if X is a supe rreflexive and hereditarily indecomposable space. On the other hand, if X i s superreflexive and X has a complemented subspace with unconditional basis or, more generally, if X has a polynomially bounded and not superstable op erator, then the set S(X) is non Borel.