Homogeneous $2$-shifts

TL;DR

This paper classifies all irreducible homogeneous bi-lateral 2-by-2 block shifts by constructing new examples and proving they, along with previous ones, exhaust all such shifts up to unitary equivalence.

Contribution

The paper introduces two new classes of irreducible homogeneous bi-lateral 2-by-2 block shifts and completes their classification.

Findings

New classes of homogeneous bi-lateral 2-by-2 block shifts constructed.

Complete classification of irreducible homogeneous bi-lateral 2-by-2 block shifts achieved.

Abstract

The classification of homogeneous scalar weighted shifts is known. Recently, Kor\'{a}nyi obtained a large class of inequivalent irreducible homogeneous bi-lateral $2$-by-$2$ block shifts. In this paper, we construct two distinct classes of examples not in the list of Kor\'{a}nyi. It is then shown that these new examples of irreducible homogeneous bi-lateral $2$-by-$2$ block shifts, together with the ones found earlier by Kor\'{a}nyi, account for every unitarily inequivalent irreducible homogeneous bi-lateral $2$-by-$2$ block shift.