Subproduct systems of Hilbert spaces: dimension two

TL;DR

This paper classifies all subproduct systems of two-dimensional Hilbert spaces and shows they can only generate type I1 Arveson systems, addressing a specific case in the broader classification problem.

Contribution

It provides a complete classification of subproduct systems of 2D Hilbert spaces and confirms they only generate type I1 Arveson systems, answering a specific case of Bhat's question.

Findings

All 2D subproduct systems generate type I1 Arveson systems.

Classification of subproduct systems up to isomorphism.

Open problem remains for higher dimensions (n=3,4,...)

Abstract

A subproduct system of two-dimensional Hilbert spaces can generate an Arveson system of type I1 only. All possible cases are classified up to isomorphism. This work is triggered by a question of Bhat: can a subproduct system of n-dimensional Hilbert spaces generate an Arveson system of type II or III? The question is still open for n=3,4,...