Holomorphic mappings of domains in operator spaces

Throughout, most of the domains we have considered are circular domains in the sense of [13]. Specifically, if 2Θ is a closed complex subspace of of ℒ(H, K), a circular domain in 2Θ is a set of the form D = {Z ∈ 2Θ : Z* EZ + 2 Re F* Z + G < 0}, where E ∈ ℒ(K), F ∈ ℒ(H, K), G ∈ ℒ(H) and both E and G are self adjoint. For example, 2Θ₀ is a circular domain since 2Θ₀ = {Z ∈ 2Θ : Z*Z - I < 0}. The circular domains of the complex plane are any open disc, the exterior of any open disc, any open half-plane, any punctured plane, the entire plane and the empty set. (This terminology is taken from [24, p. 57] and [16, p. 464]. A different definition, which is also referred to as "circled," is given in [19, p. 113] and [2, p. 104].) One of our main objectives has been to determine the automorphisms of circular domains and to discover circular domains that are homogeneous. Linear fractional transformations have served as a basic tool.