[Paper Review] Schur-class multipliers on the Fock space: de Branges-Rovnyak reproducing kernel spaces and transfer-function realizations

Joseph A. Ball, Vladimir Bolotnikov+1 more|ArXiv.org|Oct 20, 2006

Holomorphic and Operator TheoryMathematics30 references19 citations

TL;DR

This paper introduces a noncommutative Fock-space analogue of de Branges-Rovnyak reproducing kernel Hilbert spaces and establishes that every Schur-class multiplier on the Fock space admits a unique observable, coisometric transfer-function realization in the state space $\mathcal{H}(K_S)$. The key contribution is a realization-theoretic characterization of inner Schur-class multipliers and a calculus for constructing such multipliers with prescribed left zero-structure, with results symmetrically formulated in both left and right versions, closely mirroring the classical univariate case.

ABSTRACT

We introduce and study a Fock-space noncommutative analogue of reproducing kernel Hilbert spaces of de Branges-Rovnyak type. Results include: use of the de Branges-Rovnyak space ${\mathcal H}(K_{S})$ as the state space for the unique (up to unitary equivalence) observable, coisometric transfer-function realization of the Schur-class multiplier $S$, realization-theoretic characterization of inner Schur-class multipliers, and a calculus for obtaining a realization for an inner multiplier with prescribed left zero-structure. In contrast with the parallel theory for the Arveson space on the unit ball ${\mathbb B}^{d} \subset {\mathbb C}^{d}$ (which can be viewed as the symmetrized version of the Fock space used here), the results here are much more in line with the classical univariate case, with the extra ingredient of the existence of all results having both a ``left'' and a ``right'' version.

Motivation & Objective

• To develop a noncommutative analogue of de Branges-Rovnyak reproducing kernel Hilbert spaces in the Fock space setting.
• To characterize Schur-class multipliers on the Fock space via observable, coisometric transfer-function realizations.
• To provide a realization-theoretic characterization of inner Schur-class multipliers in the noncommutative Fock space framework.
• To construct a calculus for realizing inner multipliers with prescribed left zero-structure in this setting.

Proposed method

• The authors define the de Branges-Rovnyak space $\mathcal{H}(K_S)$ as the state space for a transfer-function realization of a Schur-class multiplier $S$ on the Fock space.
• They use the coisometric transfer-function realization form $S(z) = D + C(I - Z(z)A)^{-1}Z(z)B$ with a unitary colligation $\mathbf{U} = \begin{bmatrix} A & B \\ C & D \end{bmatrix}$.
• The realization is constructed via an output pair $(C, \mathbf{A})$ derived from the input pair $({\mathbf{Z}}, X)$, where $\mathbf{Z}^*$ is strongly stable and $X$ is an isometry.
• A Cholesky factorization is applied to construct the operator $\begin{bmatrix} B \\ D \end{bmatrix}$ such that $\mathbf{U}$ is coisometric.
• The space $\mathcal{M}_{\mathbf{Z},X} = \ker \mathcal{C}_{\mathbf{Z},X}$ is shown to be equal to $\theta \cdot H^2_{\mathcal{U}}(\mathcal{F}_d)$ for an inner multiplier $\theta$, using the isometric inclusion of $\mathcal{H}(K_{\theta}) = \mathcal{H}(K_{C,\mathbf{A}})$.
• The duality between left and right structures is preserved throughout, with symmetric formulations for left and right realizations.

Experimental results

Research questions

• RQ1 How can de Branges-Rovnyak spaces be generalized to the noncommutative Fock space setting to support Schur-class multiplier theory?
• RQ2 What is the unique observable, coisometric transfer-function realization of a Schur-class multiplier on the Fock space?
• RQ3 How can inner Schur-class multipliers be characterized using realization theory in this noncommutative context?
• RQ4 Can a calculus be developed to construct inner multipliers with a prescribed left zero-structure in the Fock space setting?
• RQ5 Why do the results in the Fock space case resemble the classical univariate case more closely than in the Arveson space setting?

Key findings

• The de Branges-Rovnyak space $\mathcal{H}(K_S)$ serves as the state space for the unique observable, coisometric transfer-function realization of any Schur-class multiplier $S$ on the Fock space.
• Every Schur-class multiplier $S$ admits a transfer-function realization $S(z) = D + C(I - Z(z)A)^{-1}Z(z)B$ with a coisometric colligation $\mathbf{U} = \begin{bmatrix} A & B \\ C & D \end{bmatrix}$.
• An inner Schur-class multiplier $\theta$ is realized as the transfer function of a coisometric colligation, and $\theta \cdot H^2_{\mathcal{U}}(\mathcal{F}_d) = \ker \mathcal{C}_{\mathbf{Z},X}$ for an admissible input pair $({\mathbf{Z}}, X)$.
• The realization of an inner multiplier with prescribed left zero-structure is achieved via a Cholesky factorization of the defect operator $I - \begin{bmatrix} A \\ C \end{bmatrix}\begin{bmatrix} A^* & C^* \end{bmatrix}$.
• The space $\mathcal{H}(K_{\theta})$ is isometrically equal to $\mathcal{H}(K_{C,\mathbf{A}})$, and this space coincides with $\operatorname{Ran} \mathcal{O}_{C,\mathbf{A}}$, which is isometrically included in $H^2_{\mathcal{Y}}(\mathcal{F}_d)$.
• The results are symmetric in left and right formulations, reflecting a duality that closely parallels the classical univariate case, unlike the Arveson space setting.