Cascade Connections of Linear Systems and Factorizations of Holomorphic Operator Functions Around a Multiple Zero in Several Variables

TL;DR

This paper investigates the factorization of holomorphic operator-valued functions with multiple zeros in several variables, linking the problem to cascade decompositions of multiparametric linear systems and invariant subspace criteria.

Contribution

It introduces conditions for factorization of holomorphic functions with zeros at the origin, especially within the Agler--Schur class on the polydisk, via cascade decompositions of linear systems.

Findings

Factorization solvability linked to cascade decompositions.

Reduction of factorization problem to invariant subspace existence.

Criteria established for multiparametric linear system realizations.

Abstract

We show that the factorization problem $\theta (z)=\theta_2(z)\theta_1(z)$ is solvable in the class of Hilbert space operator-valued functions holomorphic on some neighbourhood of $z=0$ in $\nspace{C}{N}$ and having a zero at $z=0$ (here $\theta (z)$ has a multiple zero at $z=0$). Such a factorization problem becomes more complicated if we demand for $\theta (z), \theta_1(z)$ and $\theta_2(z)$ to be Agler--Schur-class functions on the polydisk $\nspace{D}{N}$ and for the factorization identity to hold in $\nspace{D}{N}$. In this case we reduce it to the problem on the existence of a cascade decomposition for certain multiparametric linear system $\alpha $--a conservative realization of $\theta (z)$, and give the criterion for its solvability in terms of common invariant subspaces for the $N$-tuple of main operators of $\alpha $.