Linear Fractional Transformations of Circular Domains in Operator Spaces

0. Introduction Our object is to study domains which are the region of negative definiteness of an operator-valued Hermitian form defined on a space of operators and to investigate the biholomorphic linear fractional transformations between them. This is a unified setting in which to consider operator balls, operator half-planes, strictly J-contractive operators, strictly J-dissipative operators, etc., and the biholomorphic images of these domains under linear fractional transformations. Our approach is close in spirit to that of Potapov [28], Krein and Smuljan [27] and Smuljan [33]. At the same time, because we consider subspaces of operators, our circular domains include the matrix balls which E. Cartan [6] obtained as the classical bounded symmetric domains and they include the Siegel domains of genus 2 and 3 which Pyatetskii-Shapiro [29] associates with these domains as well as the infinite