Papers by Daniel Aron Alpay
Linear Algebra and its Applications, 1994
Let K be a nonnegative matrix satisfying a matrix equation of the form A KA*-B K B* = M*JM (where... more Let K be a nonnegative matrix satisfying a matrix equation of the form A KA*-B K B* = M*JM (where A, B, and M are of a special form). We look for representations of K of the form K = /,X(y) dl(y) X(y)* where I, is a simple closed contour, d1 is a positive matrix valued measure, and the matrix valued function X (depending on A, B, and M) defines a space of functions invariant with respect to a shift associated to I,,. The method used is that of the fundamental matrix inequality of Potapov, suitably adapted to the present framework. There exists an observable pair of matrices (C, A) E Crx (m+1) X Cm+ ljxcrn+ '1 such that J is spanned by the columns {fO,.. . , f,> of C(L+ 1-AA)-'. Let dp be a @(m+l)X(m+l)-valued finite and positive LZNEARALGEBRAANDITSAPPLICATIONS 203-204~3-43 (1994) 3
A Schur algorithm for transfer function of over-determined conservative systems, invariant in one direction
Comptes Rendus Mathematique, 2009
A generalization of the Wick-Itô stochastic integral
Comptes Rendus Mathematique, 2008
Linear Algebra and its Applications, 2006
We construct coisometric and quasi-coisometric realizations for transfer operators of multiscale ... more We construct coisometric and quasi-coisometric realizations for transfer operators of multiscale causal stationary dissipative systems.
A Panorama of Modern Operator Theory and Related Topics, 2012
It was recently shown that the theory of linear stochastic systems can be viewed as a particular ... more It was recently shown that the theory of linear stochastic systems can be viewed as a particular case of the theory of linear systems on a certain commutative ring of power series in a countable number of variables. In the present work we study an interpolation problem in this setting. A key tool is the principle of permanence of algebraic identities. Contents 14 6. More general interpolation problem 17 References 18 1991 Mathematics Subject Classification. 60H40, 93C05. Key words and phrases. white noise space, stochastic distributions, linear systems on rings. D. Alpay thanks the Earl Katz family for endowing the chair which supported his research.
We here characterize the minimality of realization of arbitrary linear time-invariant dynamical s... more We here characterize the minimality of realization of arbitrary linear time-invariant dynamical systems through (i) intersection of the spectra of the realization matrix and of the corresponding state submatrix and (ii) moving the poles by applying static output feedback. In passing, we introduce, for a given square matrix A, a parameterization of all matrices B for which the pairs (A, B) are controllable. In particular, the minimal rank of such B turns to be equal to the smallest geometric multiplicity among the eigenvalues of A. Finally, we show that the use of a (not necessarily square) realization matrix L to examine minimality of realization, is equivalent to the study of a smaller dimensions, square realization matrix L_sq, which in turn is linked to realization matrices obtained as polynomials in L_sq. Namely a whole family of systems.
Operator Theory: Advances and Applications, 2005
La séduction de certains problèmes vient de leur défaut de rigueur, comme des opinions discordant... more La séduction de certains problèmes vient de leur défaut de rigueur, comme des opinions discordantes qu'ils suscitent: autant de difficultés dont s'entiche l'amateur d'Insoluble.
Representation Formulas for Hardy Space Functions Through the Cuntz Relations and New Interpolation Problems
Multiscale Signal Analysis and Modeling, 2012
Systems & Control Letters, 2010
We give a simple constructive proof of a factorization theorem for scalar rational functions with... more We give a simple constructive proof of a factorization theorem for scalar rational functions with a non-negative real part on the imaginary axis. A Mathematica program, performing this factorization, is provided.
Opuscula Mathematica, 2012
Using the white noise space setting, we define and study stochastic integrals with respect to a c... more Using the white noise space setting, we define and study stochastic integrals with respect to a class of stationary increment Gaussian processes. We focus mainly on continuous functions with values in the Kondratiev space of stochastic distributions, where use is made of the topology of nuclear spaces. We also prove an associated Ito formula.
Linear Algebra and its Applications, 2013
Scalar rational functions with a non-negative real part on the right half plane, called positive,... more Scalar rational functions with a non-negative real part on the right half plane, called positive, are classical in the study of electrical networks, dissipative systems, Nevanlinna-Pick interpolation and other areas. We here study generalized positive functions, i.e with a non-negative real part on the imaginary axis. These functions form a Convex Invertible Cone, cic in short, and we explore two partitionings of this set: (i) into (infinitely many non-invertible) convex cones of functions with prescribed poles and zeroes in the right half plane and (ii) each generalized positive function can be written as a sum of even and odd parts. The sets of even generalized positive and odd functions form subcics. It is well known that the Nevanlinna-Pick interpolation problem is not always solvable by positive functions. Unfortunately, there is no computationally simple procedure to carry out this interpolation in the framework of generalized positive functions. Through examples it is illustrated how the two above partitionings of generalized positive functions can be exploited to introduce simple ways to carry out the Nevanlinna-Pick interpolation. Finally we show that only some of these properties are carried over to rational generalized bounded functions, mapping the imaginary axis to the unit disk.
Linear Algebra and its Applications, 2007
We prove representation theorems for Carathéodory functions in the setting of Banach spaces.
Linear Algebra and its Applications, 2009
We study Carathéodory-Herglotz functions whose values are continuous operators from a locally con... more We study Carathéodory-Herglotz functions whose values are continuous operators from a locally convex topological vector space which admits the factorization property into its conjugate dual space. We show how this case can be reduced to the case of functions whose values are bounded operators from a Hilbert space into itself.
Linear Algebra and its Applications, 2003
Recently Bolotnikov and Rodman characterized finite dimensional backward shift invariant subspace... more Recently Bolotnikov and Rodman characterized finite dimensional backward shift invariant subspaces of the Arveson space. In this note we study finite dimensional backward shift invariant Hilbert spaces contractively included in the Arveson space. To that purpose we redefine in an appropriate way the backward shift operators in terms of power series expansions.
Linear Algebra and its Applications, 2014
We here specialize the standard matrix-valued polynomial interpolation to the case where on the i... more We here specialize the standard matrix-valued polynomial interpolation to the case where on the imaginary axis the interpolating polynomials admit various symmetries: Positive semidefinite, Skew-Hermitian, J-Hermitian, Hamiltonian and others. The procedure is comprized of three stages, illustrated through the case where on iR the interpolating polynomials are to be positive semidefinite. We first, on the expense of doubling the degree, obtain a minimal degree interpolating polynomial P (s) which on iR is Hermitian. Then we find all polynomials Ψ(s), vanishing at the interpolation points which are positive semidefinite on iR. Finally, using the fact that the set of positive semidefinite matrices is a convex subcone of Hermitian matrices, one can compute the minimal scalarβ ≥ 0 so that P (s)+βΨ(s) satisfies all interpolation constraints for all β ≥β. This approach is then adapted to cases when the family of interpolating polynomials is not convex. Whenever convex, we parameterize all minimal degree interpolating polynomials.
Journal of Mathematical Analysis and Applications, 2002
We solve Gleason's problem in the reproducing kernel Hilbert spaces with reproducing kernels 1/(1... more We solve Gleason's problem in the reproducing kernel Hilbert spaces with reproducing kernels 1/(1 − N 1 z j w j) r for real r > 0 and their counterparts for r 0, and study the homogeneous interpolation problem in these spaces.
Journal of Functional Analysis, 2005
We introduce the notion of rationality for hyperholomorphic functions (functions in the kernel of... more We introduce the notion of rationality for hyperholomorphic functions (functions in the kernel of the Cauchy-Fueter operator). Following the case of one complex variable, we give three equivalent definitions: the first in terms of Cauchy-Kovalevskaya quotients of polynomials, the second in terms of realizations and the third in terms of backward-shift invariance. Also introduced and studied are the counterparts of the Arveson space and Blaschke factors.
Journal of Functional Analysis, 2009
The classical concept of Q-functions associated to symmetric and selfadjoint operators due to M.G... more The classical concept of Q-functions associated to symmetric and selfadjoint operators due to M.G. Krein and H. Langer is extended in such a way that the Dirichlet-to-Neumann map in the theory of elliptic differential equations can be interpreted as a generalized Q-function. For couplings of uniformly elliptic second order differential expression on bounded and unbounded domains explicit Krein type formulas for the difference of the resolvents and trace formulas in an H 2-framework are obtained.
Journal of Functional Analysis, 2002
We study certain finite dimensional reproducing kernel indefinite inner product spaces of multipl... more We study certain finite dimensional reproducing kernel indefinite inner product spaces of multiplicative half order differentials on a compact real Riemann surface; these spaces are analogues of the spaces introduced by L. de Branges when the Riemann sphere is replaced by a compact real Riemann surface of a higher genus. In de Branges theory an important role is played by resolvent-like difference quotient operators R : ; here we introduce generalized difference quotient operators R y : for any non-constant meromorphic function y on the Riemann surface. The spaces we study are invariant under generalized difference quotient operators and can be characterized as finite dimensional indefinite inner product spaces invariant under two operators R y 1 : i and R y 2 : 2 , where y 1 and y 2 generate the field of meromorphic functions on the Riemann surface, which satisfy a supplementary identity, analogous to the de Branges identity for difference quotients. Just as the classical de Branges spaces and difference quotient operators appear in the operator model theory for a single nonselfadjoint (or nonunitary) operator, the spaces we consider and generalized difference quotient operators appear in the model theory for commuting nonselfadjoint operators with finite nonhermitian ranks.
Journal of Functional Analysis, 2003
We prove a trace formula for pairs of self-adjoint operators associated to canonical differential... more We prove a trace formula for pairs of self-adjoint operators associated to canonical differential expressions. An important role is played by the associated Weyl function.
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Papers by Daniel Aron Alpay