Matrix-valued Case Research Articles

Stability and convergence of Strang splitting. Part II: Tensorial Allen-Cahn equations

We consider the second-order in time Strang-splitting approximation for tensorial (e.g. vector-valued and matrix-valued) Allen-Cahn equations. Both the linear propagator and the nonlinear propagator are computed explicitly. For the vector-valued case, we prove the maximum principle and unconditional energy dissipation for a judiciously modified energy functional. The modified energy functional is close to the classical energy up to O(τ) where τ is the splitting step. For the matrix-valued case, we prove a sharp maximum principle in the matrix Frobenius norm. We show modified energy dissipation under very mild splitting step constraints. We exhibit several numerical examples to show the efficiency of the method as well as the sharpness of the results.

Matrix-valued Aleksandrov–Clark measures and Carathéodory angular derivatives

This paper deals with families of matrix-valued Aleksandrov–Clark measures {μα}α∈U(n), corresponding to purely contractive n×n matrix functions b on the unit disc of the complex plane. We do not make other apriori assumptions on b. In particular, b may be non-inner and/or non-extreme. The study of such families is mainly motivated from applications to unitary finite rank perturbation theory.A description of the absolutely continuous parts of μα is a rather straightforward generalization of the well-known results for the scalar case (n=1).The results and proofs for the singular parts of matrix-valued μα are more complicated than in the scalar case, and constitute the main focus of this paper. We discuss matrix-valued Aronszajn–Donoghue theory concerning the singular parts of the Clark measures, as well as Carathéodory angular derivatives of matrix-valued functions and their connections with atoms of μα. These results are far from being straightforward extensions from the scalar case: new phenomena specific to the matrix-valued case appear here. New ideas, including the notion of directionality, are required in statements and proofs.

Quaternionic Wiener Algebras, Factorization and Applications

We define an almost periodic extension of the Wiener algebras in the quaternionic setting and prove a Wiener-Levy type theorem for it, as well as extending the theorem to the matrix-valued case. We prove a Wiener-Hopf factorization theorem for the quaternionic matrix-valued Wiener algebras (discrete and continuous) and explore the connection to the Riemann-Hilbert problem in that setting. As applications, we characterize solvability of two classes of quaternionic functional equations and give an explicit formula for the canonical factorization of quaternionic rational matrix functions via realization.

Adaptive orthonormal systems for matrix-valued functions

In this paper we consider functions in the Hardy space $\mathbf{H}_2^{p\times q}$ defined in the unit disc of matrix-valued. We show that it is possible, as in the scalar case, to decompose those functions as linear combinations of suitably modified matrix-valued Blaschke product, in an adaptive way. The procedure is based on a generalization to the matrix-valued case of the maximum selection principle which involves not only selections of suitable points in the unit disc but also suitable orthogonal projections. We show that the maximum selection principle gives rise to a convergent algorithm. Finally, we discuss the case of real-valued signals.

Stochastic differential equations for models of non-relativistic matter interacting with quantized radiation fields

We discuss Hilbert space-valued stochastic differential equations associated with the heat semi-groups of the standard model of non-relativistic quantum electrodynamics and of corresponding fiber Hamiltonians for translation invariant systems. In particular, we prove the existence of a stochastic flow satisfying the strong Markov property and the Feller property. To this end we employ an explicit solution ansatz. In the matrix-valued case, i.e., if the electron spin is taken into account, it is given by a series of operator-valued time-ordered integrals, whose integrands are factorized into annihilation, preservation, creation, and scalar parts. The Feynman–Kac formula implied by these results is new in the matrix-valued case. Furthermore, we discuss stochastic differential equations and Feynman–Kac representations for an operator-valued integral kernel of the semi-group. As a byproduct we obtain analogous results for Nelson’s model.

Convergence Analysis for Splitting of the Abstract Differential Riccati Equation

We consider a splitting-based approximation of the abstract differential Riccati equation in the setting of Hilbert--Schmidt operators. The Riccati equation arises in many different areas and is important within the field of optimal control. In this paper we conduct a temporal error analysis and prove that the splitting method converges with the same order as the implicit Euler scheme, under the same low regularity requirements on the initial values. For a subsequent spatial discretization, the abstract setting also yields uniform temporal error bounds with respect to the spatial discretization parameter. The spatial discretizations commonly lead to large-scale problems, where the use of structural properties of the solution is essential. We therefore conclude by proving that the splitting method preserves low-rank structure in the matrix-valued case. Numerical results demonstrate the validity of the convergence analysis.

Rational Cayley inner Herglotz–Agler functions: Positive-kernel decompositions and transfer-function realizations

The Bessmertnyĭ class consists of rational matrix-valued functions of d complex variables representable as the Schur complement of a block of a linear pencil A(z)=z1A1+⋯+zdAd whose coefficients Ak are positive semidefinite matrices. We show that it coincides with the subclass of rational functions in the Herglotz–Agler class over the right poly-halfplane which are homogeneous of degree one and which are Cayley inner. The latter means that such a function is holomorphic on the right poly-halfplane and takes skew-Hermitian matrix values on (iR)d, or equivalently, is the double Cayley transform (over the variables and over the matrix values) of an inner function on the unit polydisk. Using Agler–Knese's characterization of rational inner Schur–Agler functions on the polydisk, extended now to the matrix-valued case, and applying appropriate Cayley transformations, we obtain characterizations of matrix-valued rational Cayley inner Herglotz–Agler functions both in the setting of the polydisk and of the right poly-halfplane, in terms of transfer-function realizations and in terms of positive-kernel decompositions. In particular, we extend Bessmertnyĭ's representation to rational Cayley inner Herglotz–Agler functions on the right poly-halfplane, where a linear pencil A(z) is now in the form A(z)=A0+z1A1+⋯+zdAd with A0 skew-Hermitian and the other coefficients Ak positive semidefinite matrices.

A note on the Fredholm properties of Toeplitz operators on weighted Bergman spaces with matrix-valued symbols

We characterize the essential spectra of Toeplitz operators Ta on weighted Bergman spaces with matrix-valued symbols; in particular we deal with two classes of symbols, the Douglas algebra C+H∞ and the Zhu class Q := L∞ ∩VMO∂ . In addition, for symbols in C+H∞ , we derive a formula for the index of Ta in terms of its symbol a in the scalar-valued case, while in the matrix-valued case we indicate that the standard reduction to the scalar-valued case fails to work analogously to the Hardy space case. Mathematics subject classification (2010): 47B35,

A Matrix-Valued Aleksandrov Disintegration Theorem

The system of measures constructed by Clark (J Anal Math 25:169–191, 1972), and studied extensively by Aleksandrov and others provide useful information about the properties of analytic self-maps of the unit disc. In their paper, Kapustin and Poltoratski (J Funct Anal 238:313–326, 2006) extended the concept to include operator (and hence matrix) valued analytic functions, though they restrict their attention to the interesting special case of operator-valued inner functions. In this paper, we present an alternative (though equivalent) construction, which applies only to the matrix-valued case, though not only to inner functions. This construction provides us with interesting analogues of certain theorems known to be true in the scalar case. In particular, we present a proof of a matrix-valued generalisation of the celebrated Aleksandrov Disintegration Theorem (Inv Akad Nauk ArmSSR Mat 22:490–503, 1987) to a certain class of matrix-valued functions.

Discrete Systems and their Characteristic Spectral Functions

We define first-order discrete systems in the matrix-valued case. They are characterized by sequences of pair of matrices, called admissible sequences. We present two important examples of such sequences, called Szego and Nehari sequences. We introduce the characteristic spectral functions associated to a first-order system. We define in particular the scattering function, the Weyl function and the reflection coefficient function and we study the relationships between these functions.

Matrix-valued nevanlinna-pick interpolation with complexity constraint: an optimization approach

Over the last several years, a new theory of Nevanlinna-Pick interpolation with complexity constraint has been developed for scalar interpolants. In this paper we generalize this theory to the matrix-valued case, also allowing for multiple interpolation points. We parameterize a class of consisting of most interpolants of no higher degree than the central solution in terms of spectral zeros. This is a complete parameterization, and for each choice of interpolant we provide a convex optimization problem for determining it. This is derived in the context of duality theory of mathematical programming. To solve the convex optimization problem, we employ a homotopy continuation technique previously developed for the scalar case. These results can be applied to many classes of engineering problems, and, to illustrate this, we provide some examples. In particular, we apply our method to a benchmark problem in multivariate robust control. By constructing a controller satisfying all design specifications but having only half the McMillan degree of conventional H/sup /spl infin// controllers, we demonstrate the advantage of the proposed method.

The Carathéodory–Toeplitz problem with partial data

We treat a positive extension problem in an algebra of periodic doubly infinite operator matrices. The given data have a stalactite-type pattern. The problem may be viewed as an operator-valued Carathéodory–Toeplitz problem with partially described data. Necessary and sufficient conditions are given for the existence of a solution, the set of all solutions is described and a maximum entropy result is obtained as well. In the matrix-valued case the problem was also considered by Alpay et al. [Linear Algebra Appl. 268 (1998) 247], where the problem was rephrased as an interpolation problem requiring symmetries and solutions were given in these terms. We shall also interpret our results in the context of cyclostationary stochastic processes.

On Superoptimal Approximation by Analytic and Meromorphic Matrix-Valued Functions

It is a well-known fact that for any continuous scalar-valued function φ on the unit circle there is a unique best approximation in the Hardy class H ∞ of bounded analytic functions. However, in the matrix-valued case a best approximation by bounded analytic functions is almost never unique. To make it unique, one has to impose additional assumptions. In 1986 N. J. Young introduced the so-called superoptimal solution of the Nehari problem. The word superoptimal means that we are seeking F ∈ H ∞ (matrix-valued) to minimize not only sup {|| Φ (ζ) − F(ζ) || : |ζ| = 1} = sup{ s 0(Φ(ζ) − F (ζ)) : |ζ| = 1} but also the suprema of further singular values s j (Φ(ζ) − F(ζ)), j ≥ 1. It was proved by V. V. Peller and N. J. Young that for matrix-valued function Φ in H ∞ + C such superoptiminal solution is unique. In this paper an alternative geometric approach to the problem is presented. It allows us to obtain another proof of the uniqueness and some new results: uniqueness of the superoptimal approximation by meromorphic functions (a generalization of the Adamyan-Arov-Krein result on approximation of scalar-valued functions), inequalities between s-numbers of a Hankel operator, and superoptimal singular values. Our approach works well for operator-valued functions as for matrix-valued ones.

Computing matrix-valued Nevanlinna-Pick interpolation

We describe a computational method, known as the Nevanlinna algorithm, for the matrix-valued Nevanlinna-Pick interpolation. The original interpolation problem formulated using the Carathéodory class of matrix-valued rational functions is first converted to an equivalent setting using the Schur class of rational functions. As a result, the necessary and sufficient Pick's condition for the interpolation becomes consistent with the scalar-valued formulation, so that some efficient techniques developed for the scalar-valued interpolation can be employed or modified for the matrix-valued case. We give a brief, yet sufficiently clear, derivation and a detailed arithmetic complexity analysis for the algorithm. We show that an n-point matrix-valued Nevanlinna-Pick interpolation using the new algorithm requires approximately 95 nm 3 complex arithmetic operations, where m is the matrix dimension.

Extension theorems for Invertibility Symbols in Banach algebras

Necessary and sufficient conditions are found for the possibility of an extension of Invertibility Symbols from a dense subalgebra to its closure. It is proved that such an extension is possible if and only if the closure does not have any Kakutani elements. The extension of Invertibility Symbols from a Banach algebra to the matrix-valued case is also considered. Algebras with Polynomial Identities play an important role here. Applications to algebras generated by two idempotents and to algebras generated by singular integral operators are suggested.

On a theorem of Hermite and Hurwitz

The Hermite-Hurwitz theorem computes the degree, over R , of a real rational function ƒ in terms of the signature of an associated quadratic form—known today as the Hankel matrix of ƒ. This formula, which Hermite was led to by his work on the problem of representing integers as sums of squares, gave rise to striking applications in the theory of equations and in the stability theory of ordinary differential equations. In this paper, this theorem and various generalizations to the matrix-valued case are discussed and described in terms of signature formulae. These include its relation to stability theory and the matrix Hermite-Hurwitz theorem of Bitmead-Anderson as applied to questions of circuit synthesis. This also includes a global form of Hörmander's signature formula for the Maslov index of a rational loop in a Lagrangian Grassmannian, due to Byrnes and Duncan, and applications to the topology of spaces of rational matrix-valued functions, following the work of Brockett, Byrnes, and Duncan. This includes, in particular, a topological proof of the matrix Hermite-Hurwitz theorem.