Kreĭn space representation and Lorentz groups of analytic Hilbert modules

Abstract

This paper aims to introduce some new ideas into the study of submodules in Hilbert spaces of analytic functions. The effort is laid out in the Hardy space over the bidisk $H^2({\mathbb~D}^2)$. A closed subspace $M$ in $H^2({\mathbb~D}^2)$ is called asubmodule if $z_{i}M\subset~M$ $(i=1,~2)$. An associated integral operator (em defect operator $C_{M}$ captures much information about $M$. Using a Kreuin space indefinite metric on the range of $C_{M}$, this paper gives a representation of $M$. Then it studies the group (called Lorentz group) of isometric self-maps of $M$ with respect to the indefinite metric, and in finite rank case shows that the Lorentz group is a complete invariant for congruence relation. Furthermore, the Lorentz group contains an interesting abelian subgroup (called little Lorentz group) which turns out to be a finer invariant for $M$.