Unbounded Symmetric Homogeneous Domains in Spaces of Operators

We define the domain of a linear fractional transformation in a space of operators and show that both the affine automorphisms and the compositions of symmetries act transitively on these domains. Further, we show that Liouville’s theorem holds for domains of linear fractional transformations and, with an additional trace class condition, so does the Riemann removable singularities theorem. We also show that every biholomorphic mapping of the operator domain I < ZZ is a linear isometry when the space of operators is a complex Jordan subalgebra of L(H) with the removable singularity property and that every biholomorphic mapping of the operator domain I +Z 1Z1 < Z ∗ 2Z2 is a linear map obtained by multiplication on the left and right by J-unitary and unitary operators, respectively. 0. Introduction. This paper introduces a large class of finite and infinite dimensional symmetric affinely homogeneous domains which are not holomorphically equivalent to any bounded domain. These domains are subsets of spaces of operators and include domains as diverse as a closed complex subspace of the bounded linear operators from one Hilbert space to another, the identity component of the group of invertible operators in a C-algebra and the complement of a hyperplane in a Hilbert space. Each of our domains may be characterized as a component of the domain of definition of some linear fractional transformation which maps a neighborhood of a point in the component biholomorphically onto an open set in the same space. Thus we refer to our domains as domains of linear fractional transformations. We show that at any point of such a domain, there exists a biholomorphic linear fractional transformation of the domain onto itself which is a symmetry at the point. This linear fractional transformation is a generalization of the Potapov-Ginzburg transformation. Moreover, for any two points Z0 and W0 in the domain, there exists an affine automorphism of the domain of the form φ(Z) = W0+A(Z−Z0)B and φ is a composition of the above symmetries.