A Wolff Theorem for finite rank bounded symmetric domains

Abstract

We present a Wolff Theorem for all infinite dimensional bounded symmetric domains of finite rank. Namely, if B is the open unit ball of any finite rank JB *triple and f : B → B is a compact holomorphic map with no fixed point in B, we prove convex f-invariant subdomains of B (of all sizes and at all points) exist in the form of simple operator balls c λ + T λ (B), for c λ ∈ B and T λ an invertible linear map. These are exact infinite dimensional analogues of the invariant discs in ∆, the invariant ellipsoids in the Hilbert ball and invariant domains in finite dimensional triples. Results are new for rank > 2, even for classical spaces such as C *-algebras and JB *-algebras.