[Paper Review] Abstract interpolation problem in generalized Schur classes

Dmytro Baidiuk|arXiv (Cornell University)|Mar 17, 2014

Holomorphic and Operator Theory6 references3 citations

TL;DR

This paper introduces an indefinite variant of the abstract interpolation problem in generalized Schur classes, reducing it to the extension problem of a model Pontryagin space isometric operator $V$. It establishes a one-to-one correspondence between solutions and a subclass of $\mathfrak{L}$-regular unitary extensions $U$ of $V$, realized as unitary colligations over the indefinite de Branges-Rovnyak space $\mathcal{D}(s)$, providing a complete parametrization of solutions via characteristic functions.

ABSTRACT

An indefinite variant of the abstract interpolation problem is considered. Associated to this problem is a model Pontryagin space isometric operator V. All the solutions of the problem are shown to be in a one-to-one correspondence with a subset of the set of all unitary extenions U of V. These unitary extension U of V are realized as unitary colligations with the indefinite de Branges-Rovnyak space as a state space.

Motivation & Objective

• To extend the abstract interpolation framework to indefinite settings using generalized Schur classes.
• To characterize solutions of the abstract interpolation problem $AIP(\kappa)$ in terms of unitary extensions of an isometric operator $V$ in a Pontryagin space.
• To identify a subclass of $\mathfrak{L}$-regular unitary extensions that parametrize all solutions of $AIP(\kappa)$.
• To realize these unitary extensions as unitary colligations with the indefinite de Branges-Rovnyak space $\mathcal{D}(s)$ as the state space.
• To provide a complete parametrization of solutions using characteristic functions and operator-theoretic constructions.

Proposed method

• Reduces the indefinite abstract interpolation problem $AIP(\kappa)$ to the extension problem of a model isometric operator $V$ in a Pontryagin space.
• Identifies solutions via $\mathfrak{L}$-regular unitary extensions $U$ of $V$, which are characterized by a kernel condition involving $C(M - \lambda N)^{-1}$.
• Realizes the unitary extension $U$ as a unitary colligation $\Delta = (\mathcal{H}, \mathfrak{L}_2, \mathfrak{L}_1; U)$ with state space $\mathcal{D}(s)$, the indefinite de Branges-Rovnyak space.
• Uses the characteristic function $s(\lambda)$ of the colligation to parametrize solutions via $s(\lambda) = (w_{11}\varepsilon + w_{12})(w_{21}\varepsilon + w_{22})^{-1}$, where $\varepsilon \in S(\mathfrak{L}_2, \mathfrak{L}_1)$.
• Applies results from unitary colligation theory and reproducing kernel spaces to ensure the solution set is fully described.
• Employs the $J$-unitary property of $W(\lambda)$ on the unit circle and spectral conditions to ensure $s(\lambda) \in \mathcal{P}_\kappa(J)$.

Experimental results

Research questions

• RQ1 How can the classical abstract interpolation problem in the Schur class be generalized to an indefinite setting with a negative index $\kappa$?
• RQ2 What class of unitary extensions of the model isometric operator $V$ corresponds exactly to the solutions of the indefinite $AIP(\kappa)$?
• RQ3 How can the solution set of $AIP(\kappa)$ be parametrized in terms of operator colligations and characteristic functions in a Pontryagin space?
• RQ4 What role does the indefinite de Branges-Rovnyak space $\mathcal{D}(s)$ play in realizing the unitary colligation and characterizing solutions?
• RQ5 Under what conditions does the characteristic function $W(\lambda)$ of the colligation ensure that $s(\lambda) \in \mathcal{U}_\kappa(J)$?

Key findings

• Solutions of the $AIP(\kappa)$ problem are in one-to-one correspondence with $\mathfrak{L}$-regular unitary extensions $U$ of the model isometric operator $V$.
• The solution set is parametrized by $s(\lambda) = (w_{11}\varepsilon + w_{12})(w_{21}\varepsilon + w_{22})^{-1}$, where $\varepsilon$ ranges over $S(\mathfrak{L}_2, \mathfrak{L}_1)$ and $w_{21}(0)\varepsilon(0) + w_{22}(0)$ is invertible.
• The mapping $\Phi: \mathcal{H} \to \mathcal{D}(s)$ is uniquely defined by $\Phi(t) = \begin{bmatrix} I & -s(t) \ -s^*(t) & I \end{bmatrix} C(M - tN)^{-1}$ for $t \in \mathbb{T}$.
• The characteristic function $s(\lambda)$ belongs to $\mathcal{P}_\kappa(J)$ if and only if $\bigcap_{\lambda \in \rho(M,N)} \ker C(M - \lambda N)^{-1} = \{0\}$.
• If $m(\sigma(M,N) \cap \mathbb{T}) = 0$, then $W(\lambda)$ is $J$-unitary a.e. on $\mathbb{T}$, ensuring $s(\lambda) \in \mathcal{U}_\kappa(J)$.
• The solution $s(\lambda)$ is holomorphic in a neighborhood of $0$ and lies in $S_\kappa(\mathfrak{L}_2, \mathfrak{L}_1)$, with the unitary operator $U$ being $({\mathfrak{L}}_2,{\mathfrak{L}}_1)$-regular.