Finite Rank Isopairs

TL;DR

This paper introduces the concept of rank for algebraic isopairs, characterizing their structure via restrictions to invariant subspaces and providing a detailed algebraic framework for finite bimultiplicity cases.

Contribution

It defines the rank of pure algebraic isopairs with finite bimultiplicity and describes their structure as restrictions of cyclic isopairs to finite codimensional invariant subspaces.

Findings

Defined the rank as an s-tuple for algebraic isopairs

Characterized isopairs of finite bimultiplicity as restrictions of cyclic isopairs

Established properties of restrictions to invariant subspaces

Abstract

An algebraic isopair is a commuting pair of pure isometries that is annihilated by a polynomial defining a distinguished variety $\mathcal{V}$. The notion of the rank of a pure algebraic isopair with finite bimultiplicity is introduced. For $\mathcal{V} $, a union of $s$ irreducible varieties $\mathcal{V}_j$, the rank is a $s$-tuple $\alpha=(\alpha_1,...,\alpha_s)$ of natural numbers. A pure algebraic isopair of finite bimultiplicity with rank $\alpha$ is described as a restriction of a $\max\{\alpha_1,...,\alpha_s\}$-cyclic pure algebraic isopair to a finite codimensional invariant subspace. The restriction of a pure algebraic isopair of finite bimultiplicity with rank $\alpha$ to a finite codimensional invariant subspace is at least $\max\{\alpha_1,...,\alpha_s\}$-cyclic and there is a $\max\{\alpha_1,...,\alpha_s\}$-cyclic finite codimensional invariant subspace.