arXiv subjects

Sanne ter Horst

Publications and source records attributed to Sanne ter Horst.

At least 19 recordsLinked to original sources

arXiv2025

Canonical Wiener-Hopf factorization on the unit circle: matching subspaces versus Riccati equations

Wiener-Hopf factorization is an important tool in the study of block Toeplitz and block Wiener-Hopf operators, and many applications involving these operators. In this paper we compare two approaches to Wiener-Hopf factorization, namely, the more classical approach based on matching invariant subspaces and a more recent approach based on solutions to a non-symmetric Riccati equation. The latter approach is extended to the case of Hilbert space operator-valued functions that are analytic on a neighborhood of the unit disc $\BT$, but need not be rational. In both approaches, existence of canonical right Wiener-Hopf factorization is characterized by existence of a stabilizing solution to a Riccati equation, however, the Riccati equations are not the same. We analyse the solution sets of the Riccati equations and show that they indeed are not the same, but they do have the same stabilizing solution.

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arXiv2025

A new systems theory perspective on canonical Wiener-Hopf factorization on the unit circle

We establish left and right canonical factorizations of Hilbert-space operator-valued functions G(z) that are analytic on neighborhoods of the complex unit circle and the origin 0, and that have the form G(z)=I+F(z) with F(z) taking strictly contractive values on the unit circle. Such functions can be realized as transfer functions of infinite dimensional dichotomous discrete-time linear systems, and we employ the strict bounded real lemma for this class of systems, together with associated Krein space theory, to derive explicit formulas for the left and right canonical factorizations.

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arXiv2024

Optimal interpolation in Hardy and Bergman spaces: a reproducing kernel Banach space approach

After a review of the reproducing kernel Banach space framework and semi-inner products, we apply the techniques to the setting of Hardy spaces $H^p$ and Bergman spaces $A^p$, $1<p<\infty$, on the unit ball in $\mathbb{C}^n$, as well as the Hardy space on the polydisk and half-space. In particular, we show how the framework leads to a procedure to find a minimal norm element $f$ satisfying interpolation conditions $f(z_j)=w_j$, $j=1,\ldots , n$. We also explain the techniques in the setting of $\ell^p$ spaces where the norm is defined via a change of variables and provide numerical examples.

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arXiv2024

Indefinite determinantal representations versus nonsingularities on the noncommutative d-torus

We show that for a multivariable polynomial $p(z)=p(z_1, \ldots , z_d)$ with a determinantal representation $$ p(z) = p(0) \det (I_n- K (\oplus_{j=1}^d z_j I_{n_j}))$$ the matrix $K$ is structurally similar to a strictly $J$-contractive matrix for some diagonal signature matrix $J$ if and only if the extension of $p(z)$ to a polynomial in $d$-tuples of matrices of arbitrary size given by \[ p(U_1, \ldots , U_d) = p(0,\ldots,0) \det (I_n\otimes I_m- (K\otimes I_m) (\oplus_{j=1}^d I_{n_j}\otimes U_j)), \] where $U_1,\ldots , U_d \in {\mathbb C}^{m \times m}$, $m\in {\mathbb N}$, does not have roots on the noncommutative $d$-torus consisting of $d$-tuples $(U_1, \ldots , U_d)$ of unitary matrices of arbitrary size.

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arXiv2023

Equivalence after extension and Schur coupling for Fredholm operators on Banach spaces

Schur coupling (SC) and equivalence after extension (EAE) are important relations for bounded operators on Banach spaces. It has been known for 30 years that the former implies the latter, but only recently Ter Horst, Messerschmidt, Ran and Roelands disproved the converse by constructing a pair of Fredholm operators which are EAE, but not SC. Motivated by this result, we investigate when EAE and SC coincide for Fred\-holm operators. Fredholm operators which are EAE have the same Fredholm index. Surprisingly, we find that for each integer $k$ and every pair of Banach spaces $(\mathcal{X},\mathcal{Y})$, either no pair of Fredholm operators of index~$k$ acting on $\mathcal{X}$ and $\mathcal{Y}$, respectively, is SC, or every pair of this kind which is EAE is also SC. Consequently, the question whether EAE and SC coincide for Fredholm operators of index~$k$ depends only on the geometry of the underlying Banach spaces $\mathcal{X}$ and $\mathcal{Y}$, not on the properties of the operators themselves. We quantify this finding by introducing two numerical indices which capture the coincidence of EAE and SC, and provide a number of examples illustrating the possible values of these indices. Notably, this includes an example showing that the above-mentioned result of Ter Horst et al, which is based on a pair of essentially incomparable Banach spaces, does not extend to projectively incomparable Banach spaces.

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arXiv2022

Convex invertible cones and Nevanlinna-Pick interpolation: The suboptimal case

Nevanlinna-Pick interpolation developed from a topic in classical complex analysis to a useful tool for solving various problems in control theory and electrical engineering. Over the years many extensions of the original problem were considered, including extensions to different function spaces, nonstationary problems, several variable settings and interpolation with matrix and operator points. Here we discuss a variation on Nevanlinna-Pick interpolation for positive real odd functions evaluated in real matrix points. This problem was studied by Cohen and Lewkowicz using convex invertible cones and the Lyapunov order, but was never fully resolved. In this paper we present a solution to this problem in a special case that we refer to as `suboptimal' based on connections with the classical case. The solution requires a representation of linear matrix maps going back to R.D. Hill and an analysis of when positive linear matrix maps are completely positive, on which we reported in earlier work and which we will briefly review here.

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arXiv2022

The Infinite-Dimensional Standard and Strict Bounded Real Lemmas in Continuous Time: The storage function approach

The bounded real lemma (BRL) is a classical result in systems theory, which provides a linear matrix inequality criterium for dissipativity, via the Kalman-Yakubovich-Popov (KYP) inequality. The BRL has many applications, among others in H-infinity control. Extensions to infinite dimensional systems, although already present in the work of Yakubovich, have only been studied systematically in the last few decades. In this context various notions of stability, observability and controllability exist, and depending on the hypothesis one may have to allow the KYP-inequality to have unbounded solutions which forces one to consider the KYP-inequality in a spatial form. In the present paper we consider the BRL for continuous time, infinite dimensional, linear well-posed systems. Via an adaptation of Willems' storage function approach we present a unified way to address both the standard and strict forms of the BRL. We avoid making use of the Cayley transform and work only in continuous time. While for the standard bounded real lemma, we obtain analogous results as there exist for the discrete time case, when treating the strict case additional conditions are required, at least at this stage. This might be caused by the fact that the Cayley transform does not preserve exponential stability, an important property in the strict case, when transferring a continuous-time system to a discrete-time system.

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arXiv2022

Interpolation in multivariable de Branges-Rovnyak spaces

We study a general metric constrained interpolation problem in a de Branges-Rovnyak space $\mathcal{H}(K_S)$ associated with a contractive multiplier $S$ between two Fock spaces along with its commutative counterpart, a de Branges-Rovnyak space associated with a Schur multiplier on the Drury-Arveson space of the unit ball of $\mathbb{C}^n$.

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arXiv2021

A Hill-Pick matrix criteria for the Lyapunov order

The Lyapunov order appeared in the study of Nevanlinna-Pick interpolation for positive real odd functions with general (real) matrix points. For real or complex matrices $A$ and $B$ it is said that $B$ Lyapunov dominates $A$ if \begin{equation*} H=H^*,\quad HA+A^*H \geq 0 \quad \implies \quad HB+B^*H \geq 0. \end{equation*} (In case $A$ and $B$ are real we usually restrict to real Hermitian matrices $H$, i.e., symmetric $H$.) Hence $B$ Lyapunov dominates $A$ if all Lyapunov solutions of $A$ are also Lyapunov solutions of $B$. In this chapter we restrict to the case that appears in the study of Nevanlinna-Pick interpolation, namely where $B$ is in the bicommutant of $A$ and where $A$ is Lyapunov regular, meaning the eigenvalues $λ_j$ of $A$ satisfy \[ λ_i + \overlineλ_j \ne 0, \quad i,j=1,\ldots,n. \] In this case we provide a matrix criteria for Lyapunov dominance of $A$ by $B$. The result relies on a class of $*$-linear maps for which positivity and complete positivity coincide and a representation of $*$-linear matrix maps going back to work of R.D. Hill. The matrix criteria asks that a certain matrix, which we call the Hill-Pick matrix, be positive semidefinite.

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arXiv2021

Hill representations for *-linear matrix maps

In a paper from 1973 R.D. Hill studied linear matrix maps $\mathcal{L}:\mathbb{C}^{q \times q}\to\mathbb{C}^{n \times n}$ which map Hermitian matrices to Hermitian matrices, or equivalently, preserve adjoints, i.e., $\mathcal{L}(V^*)=\mathcal{L}(V)^*$, via representations of the form \begin{equation*} \mathcal{L}(V)=\sum_{k,l=1}^m \mathbb{H}_{kl}\, A_l V A_k^*,\quad V\in\mathbb{C}^{q \times q}, \end{equation*} for matrices $A_1,\ldots,A_m \in\mathbb{C}^{n \times q}$ and continued his study of such representations in later work, sometimes with co-authors, to completely positive matrix maps and associated matrix reorderings. In this paper we expand the study of such representations, referred to as Hill representations here, in various directions. In particular, we describe which matrices $A_1,\ldots, A_m$ can appear in Hill representations (provided the number $m$ is minimal) and determine the associated Hill matrix $\mathbb{H}=\left[\mathbb{H}_{kl}\right]$ explicitly. Also, we describe how different Hill representations of $\mathcal{L}$ (again with $m$ minimal) are related and investigate further the implication of $*$-linearity on the linear map $\mathcal{L}$.

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arXiv2021

Linear matrix maps for which positivity and complete positivity coincide

By the Choi matrix criteria it is easy to determine if a specific linear matrix map is completely positive, but to establish whether a linear matrix map is positive is much less straightforward. In this paper we consider classes of linear matrix maps, determined by structural conditions on an associated matrix, for which positivity and complete positivity coincide. The basis of our proofs lies in a representation of $*$-linear matrix maps going back to work of R.D. Hill which enables us to formulate a sufficient condition in terms of surjectivity of certain bilinear maps.

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arXiv2020

Realization theory for poset-causal systems: Controllability, observability and duality

Poset-causal systems form a class of decentralized systems introduced by Shah and Parrilo [32] and studied mainly in the context of optimal decentralized control. In this paper we develop part of the realization theory for poset-causal systems. More specifically, we investigate several notions of controllability and observability, and their relation under duality. These new notions extend concepts of controllability and observability in the context of coordinated linear systems [16]. While for coordinated linear systems there is a clear hierarchical structure with a single (main) coordinator, for poset-causal systems there need not be a single coordinator and the communication structure between the decentralized systems allows for more intricate structures, governed by partial orders. On the other hand, we show that the class of poset-causal systems is closed under duality, which is not the case for coordinated linear systems, and that duality relations between the various notions of observability and controllability exist.

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arXiv2020

The convex invertible cone structure of positive real odd rational matrix functions

Positive real odd matrix functions, often referred to as positive real lossless matrix functions, play an important role in many applications in multi-port electrical systems. In this paper we present closer analogues to some of the known results for the scalar, one-port, case in the multi-port setting. Specifically, we determine necessary and sufficient conditions for the well studied partial fraction formula to represent functions in the class of positive real odd matrix functions, and explicit minimal state space realization formulas for the inverse (admittance) of a function in this class, which itself is also a positive real odd matrix function. Doing so, enables us to provide a partial analogue of the poles-zero interlacing behavior from the scalar case.

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arXiv2019

Equivalence after extension and Schur coupling do not coincide, on essentially incomparable Banach spaces

In 1994 H. Bart and V.É. Tsekanovskii posed the question whether the Banach space operator relations matricial coupling (MC), equivalence after extension (EAE) and Schur coupling (SC) coincide, leaving only the implication EAE/MC $\Rightarrow$ SC open. Despite several affirmative results, in this paper we show that the answer in general is no. This follows from a complete description of EAE and SC for the case that the operators act on essentially incomparable Banach spaces, which also leads to a new characterization of the notion of essential incomparability. Concretely, the forward shift operators $U$ on $\ell^p$ and $V$ on $\ell^q$, for $1\leq p,q\leq \infty$, $p\neq q$, are EAE but not SC. As a corollary, SC is not transitive. Under mild assumptions, given $U$ and $V$ that are Atkinson or generalized invertible and EAE, we give a concrete operator $W$ that is SC to both $U$ and $V$, even if $U$ and $V$ are not SC themselves. Some further affirmative results for the case where the Banach spaces are isomorphic are also obtained.

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arXiv2018

Abstract interpolation in vector-valued de Branges-Rovnyak spaces

Following ideas from the Abstract Interpolation Problem of Katsnelson et al. (Operators in spaces of functions and problems in function theory, vol 146, pp 83-69, Naukova Dumka, Keiv, 1987) for Schur class functions, we study a general metric constrained interpolation problem for functions from a vector-valued de Branges-Rovnyak space $\mathcal{H}(K_S)$ associated with an operator-valued Schur class function $S$. A description of all solutions is obtained in terms of functions from an associated de Branges-Rovnyak space satisfying only a bound on the de Branges-Rovnyak-space norm. Attention is also paid to the case that the map which provides this description is injective. The interpolation problem studied here contains as particular cases (1) the vector-valued version of the interpolation problem with operator argument considered recently in Ball et al. (Proc Am Math Soc 139(2), 609-618, 2011) (for the nondegenerate and scalar-valued case) and (2) a boundary interpolation problem in $\mathcal{H}(K_S)$. In addition, we discuss connections with results on kernels of Toeplitz operators and nearly invariant subspaces of the backward shift operator.

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arXiv2015

Equivalence after extension for compact operators on Banach spaces

In recent years the coincidence of the operator relations equivalence after extension and Schur coupling was settled for the Hilbert space case, by showing that equivalence after extension implies equivalence after one-sided extension. In this paper we investigate consequences of equivalence after extension for compact Banach space operators. We show that generating the same operator ideal is necessary but not sufficient for two compact operators to be equivalent after extension. In analogy with the necessary and sufficient conditions on the singular values for compact Hilbert space operators that are equivalent after extension, we prove the necessity of similar relationships between the $s$-numbers of two compact Banach space operators that are equivalent after extension, for arbitrary $s$-functions. We investigate equivalence after extension for operators on $\ell^{p}$-spaces. We show that two operators that act on different $\ell^{p}$-spaces cannot be equivalent after one-sided extension. Such operators can still be equivalent after extension, for instance all invertible operators are equivalent after extension, however, if one of the two operators is compact, then they cannot be equivalent after extension. This contrasts the Hilbert space case where equivalence after one-sided extension and equivalence after extension are, in fact, identical relations. Finally, for general Banach spaces $X$ and $Y$, we investigate consequences of an operator on $X$ being equivalent after extension to a compact operator on $Y$. We show that, in this case, a closed finite codimensional subspace of $Y$ must embed into $X$, and that certain general Banach space properties must transfer from $X$ to $Y$. We also show that no operator on $X$ can be equivalent after extension to an operator on $Y$, if $X$ and $Y$ are essentially incomparable Banach spaces.

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arXiv2014

Bounded Real Lemma and structured singular value versus diagonal scaling: the free noncommutative setting

The structured singular value $μ$ was introduced independently by Doyle and Safanov as a tool for analyzing robustness of system stability and performance in the presence of structured uncertainty in the system parameters. While the structured singular value provides a necessary and sufficient criterion for robustness with respect to a structured ball of uncertainty, it is notoriously difficult to actually compute. The method of diagonal (or simply "D") scaling, on the other hand, provides an easily computable upper bound (which we call $\hat μ$) for the structured singular value, but provides an exact evaluation of $μ$ (or even a useful upper bound for $μ$) only in special cases. However it was discovered in the 1990s that a certain enhancement of the uncertainty structure (i.e., letting the uncertainty parameters be freely noncommuting linear operators on an infinite-dimensional separable Hilbert space) resulted in the $D$-scaling procedure leading to an exact evaluation of $μ_{\text{enhanced}}$ ($μ_{\text{enhanced}} = \hat μ$), at least for the tractable special cases which were analyzed in complete detail. On the one hand this enhanced uncertainty has some appeal from the physical point of view: one can allow the uncertainty in the plant parameters to be time-varying, or more generally, one can catch the uncertainty caused by the designer's decision not to model the more complex (e.g. nonlinear) dynamics of the true plant. On the other hand, the precise mathematical formulation of this enhanced uncertainty structure makes contact with developments in the growing theory of analytic functions in freely noncommuting arguments and associated formal power series in freely noncommuting indeterminates. In this article we obtain the $\widetilde μ= \hat μ$ theorem for a more satisfactory general setting.

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arXiv2013

Equivalence after extension and matricial coupling coincide with Schur coupling, on separable Hilbert spaces

It is known that two Banach space operators that are Schur coupled are also equivalent after extension, or equivalently, matricially coupled. The converse implication, that operators which are equivalent after extension or matricially coupled are also Schur coupled, was only known for Fredholm Hilbert space operators and Fredholm Banach space operators with index 0. We prove that this implication also holds for Hilbert space operators with closed range, generalizing the result for Fredholm operators, and Banach space operators that can be approximated in operator norm by invertible operators. The combination of these two results enables us to prove that the implication holds for all operators on separable Hilbert spaces.

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