Compressions of infinite-dimensional bounded symmetric domains

Abstract

If D is a bounded symmetric domain in a complex Banach space Z, then the identity component G of its group of biholomorphic automorphisms permits a natural embedding into a complex Banach—Lie group H acting partially on Z. A typical model is the action of the group PSL(2,C) by Moebius transformations. In this paper we show that the interior S ⁰ of the compression semigroup S := { h ∈ H: h.D \subeq D } has a polar decomposition in the sense that S ⁰ = G \exp(W_G^0), where W_G \subeq ig is a closed convex invariant cone and the polar map G \times W_G^0 → S^0 is a diffeomorphism.