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

Dmitriy S. Kalyuzhniy|ArXiv.org|Feb 4, 2000

Holomorphic and Operator Theory4 references3 citations

TL;DR

This paper establishes a criterion for factorizing holomorphic operator-valued functions with a multiple zero at the origin in several complex variables, specifically within the Agler–Schur class on the polydisk. It links the solvability of the factorization problem to the existence of a cascade decomposition in conservative multiparameter linear systems, showing that such a factorization exists if and only if there exists a reducing invariant subspace for the system's main operators satisfying specific spectral and duality conditions.

ABSTRACT

We show that the factorization problem $θ(z)=θ_2(z)θ_1(z)$ is solvable in the class of Hilbert space operator-valued functions holomorphic on some neighbourhood of $z=0$ in $ space{C}{N}$ and having a zero at $z=0$ (here $θ(z)$ has a multiple zero at $z=0$). Such a factorization problem becomes more complicated if we demand for $θ(z), θ_1(z)$ and $θ_2(z)$ to be Agler--Schur-class functions on the polydisk $ space{D}{N}$ and for the factorization identity to hold in $ space{D}{N}$. In this case we reduce it to the problem on the existence of a cascade decomposition for certain multiparametric linear system $α$--a conservative realization of $θ(z)$, and give the criterion for its solvability in terms of common invariant subspaces for the $N$-tuple of main operators of $α$.

Motivation & Objective

• To investigate the solvability of the factorization problem $\theta(z) = \theta_2(z)\theta_1(z)$ for operator-valued functions holomorphic near $z=0$ in $\mathbb{C}^N$ with a multiple zero at the origin.
• To extend this factorization to the Agler–Schur class on the polydisk $\mathbb{D}^N$, requiring the factorization to hold globally in the domain.
• To reduce the factorization problem to the existence of a cascade decomposition in conservative multiparameter linear systems.
• To characterize solvability in terms of common invariant subspaces for the $N$-tuple of main operators of the system.
• To establish a necessary and sufficient condition for such factorizations using spectral and duality properties of the system's unitary colligation.

Proposed method

• The paper uses conservative multiparameter linear systems $\alpha = (N; \mathbf{A}, \mathbf{B}, \mathbf{C}, \mathbf{D}; \mathcal{X}, \mathcal{U}, \mathcal{Y})$ to realize holomorphic operator functions as transfer functions $\theta_\alpha(z) = z\mathbf{D} + z\mathbf{C}(I - z\mathbf{A})^{-1}z\mathbf{B}$.
• It applies the Agler–Schur class $S_N^0(\mathcal{U}, \mathcal{Y})$, consisting of functions holomorphic on $\mathbb{D}^N$, vanishing at $z=0$, and satisfying $\|\theta(r\mathbf{T})\| \leq 1$ for all commuting contractions $\mathbf{T}$ and $r < 1$.
• The key method involves reducing the factorization problem to the existence of a cascade decomposition of a conservative system $\alpha$ into two subsystems $\alpha^{(1)}$ and $\alpha^{(2)}$, such that $\theta = \theta_{\alpha^{(2)}} \theta_{\alpha^{(1)}}$.
• It introduces the concept of a closely connected conservative realization $\alpha_{cc}$, where the state space $\mathcal{X}_{cc}$ is the closed linear span of all vectors generated by the system's operators applied to input and output spaces.
• The core technical tool is the identification of a reducing invariant subspace $\mathcal{X}_{cc}^{(2)} \subset \mathcal{X}_{cc}$ such that $\mathcal{X}_{cc}^{(2)}$ is invariant under all $A_k^{(cc)}$ and satisfies a duality condition involving the adjoint of the colligation operator $\zeta\mathbf{G}_{\alpha_{cc}}$.
• The paper proves that such a subspace exists if and only if the factorization holds in the Agler–Schur class, using spectral properties and the structure of unitary colligations on the torus $\mathbb{T}^N$.

Experimental results

Research questions

• RQ1 Under what conditions can a holomorphic operator-valued function $\theta(z)$ with a multiple zero at $z=0$ in $\mathbb{C}^N$ be factored as $\theta(z) = \theta_2(z)\theta_1(z)$ with $\theta_1, \theta_2$ in the Agler–Schur class on $\mathbb{D}^N$?
• RQ2 How is the factorization problem related to the structure of conservative multiparameter linear systems?
• RQ3 What spectral and invariant subspace conditions must be satisfied for such a factorization to exist?
• RQ4 Can the factorization be realized through a cascade decomposition of the system’s state space?
• RQ5 What role does the closely connected conservative realization play in ensuring the existence of such a factorization?

Key findings

• The factorization $\theta(z) = \theta_2(z)\theta_1(z)$ is solvable in the Agler–Schur class on $\mathbb{D}^N$ if and only if there exists a conservative realization $\alpha_{cc}$ of $\theta(z)$ with a reducing invariant subspace $\mathcal{X}_{cc}^{(2)} \subset \mathcal{X}_{cc}$ for the $N$-tuple of main operators $\mathbf{A}_{cc}$.
• The subspace $\mathcal{X}_{cc}^{(2)}$ must be invariant under all $A_k^{(cc)}$ and satisfy the duality condition $\mathcal{V}_{cc} = (\zeta\mathbf{G}_{\alpha_{cc}})^*\mathcal{Y} \ominus \mathcal{X}_{cc}^{(2)}$, where $\mathcal{V}_{cc}$ is the orthogonal complement of the image of $\mathcal{X}_{cc}^{(2)}$ under the adjoint colligation.
• The existence of such a subspace $\mathcal{X}_{cc}^{(2)}$ is both necessary and sufficient for the factorization to hold in the Agler–Schur class.
• The paper shows that even when $\mathcal{X}_{cc}^{(2)} = \{0\}$, the factorization can still be nontrivial, as the left factor may be a non-zero linear homogeneous function.
• The criterion is invariant under different constructions of $\mathcal{X}_{cc}^{(2)}$, such as $\mathcal{X}_{cc}^{(2)} = \mathcal{X}_{cc} \cap \mathcal{X}^{(2)}$, and the subspace remains valid under the same spectral and duality conditions.
• The result generalizes classical factorization theory to the multivariable setting by linking operator-theoretic factorization to the invariant subspace structure of conservative linear systems.