Loewner's theorem for maps on operator domains

TL;DR

This paper characterizes local order isomorphisms on operator domains using biholomorphic automorphisms of the generalized upper half-plane, extending classical Loewner's theorem to a broader operator setting.

Contribution

It provides an explicit description of local order isomorphisms on operator domains and explores properties of maximal local order isomorphisms, including finite-dimensional cases.

Findings

Characterization of local order isomorphisms via biholomorphic automorphisms

Explicit description of such maps on operator domains

Finite-dimensional case: order embeddings are homeomorphic order isomorphisms

Abstract

The classical Loewner's theorem states that operator monotone functions on real intervals are described by holomorphic functions on the upper half-plane. We characterize local order isomorphisms on operator domains by biholomorphic automorphisms of the generalized upper half-plane, which is the collection of all operators with positive invertible imaginary part. We describe such maps in an explicit manner, and examine properties of maximal local order isomorphisms. Moreover, in the finite-dimensional case, we prove that every order embedding of a matrix domain is a homeomorphic order isomorphism onto another matrix domain.