Finite Rank Isopairs

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.