Unbounded Symmetric Homogeneous Domains in Spaces of Operators

TL;DR

This paper explores the structure and automorphisms of unbounded symmetric homogeneous domains in operator spaces, extending classical theorems and characterizing biholomorphic mappings in these complex domains.

Contribution

It introduces new properties of operator domains, extends Liouville's and Riemann's theorems, and characterizes biholomorphic maps as linear isometries or J-unitary transformations.

Findings

Automorphisms act transitively on the domains.

Liouville's theorem extends to these operator domains.

Biholomorphic maps are characterized as linear isometries or J-unitary transformations.

Abstract

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 $I < Z^*Z$ is a linear isometry when the space of operators is a complex Jordan subalgebra of ${\cal L}(H)$ with the removable singularity property and that every biholomorphic mapping of the operator domain $I + Z_1^*Z_1 < Z_2^*Z_2$ is a linear map obtained by multiplication on the left and right by J-unitary and unitary operators, respectively. Readers interested only in the finite dimensional case may identify our spaces of operators with spaces of square and rectangular matrices.