Quasicomplete factorizations of rational matrix functions

It is shown that within the class ofn×n rational matrix functions which are analytic at infinity with valueW()=I _n, any rational matrix functionW is the productW=W ₁...W _p of rational matrix functionsW ₁,...,W _p of McMillan degree one. Furthermore, such a factorization can be established with a number of factors not exceeding 2(W)–1, where (W) denotes the McMillan degree ofW.     

On canonical factorization of rational matrix functions

This paper concerns the problem of canonical factorization of a rational matrix functionW() which is analytic but may benot invertible at infinity. The factors are obtained explicitly in terms of the realization of the original matrix function. The cases of symmetric factorization for selfadjoint and positive rational matrix functions are considered separately.     

Discrete-Time Dichotomous Well-Posed Linear Systems and Generalized Schur–Nevanlinna–Pick Interpolation

We introduce a class of matrix-valued functions W called “L²- regular”. In case W is J-inner, this class coincides with the class of “strongly regular J-inner” matrix functions in the sense of Arov–Dym. We show that the class of L²-regular matrix functions is exactly the class of transfer functions for a discrete-time dichotomous (possibly infinite-dimensional) input-state-output linear system having some additional stability properties. When applied to J-inner matrix functions, we obtain a state-space realization formula for the resolvent matrix associated with a generalized Schur–Nevanlinna–Pick interpolation problem. Communicated by Daniel Alpay Submitted: August 20, 2006; Accepted: September 13, 2006     

The functional model of a j-unitary node with a given j-inner characteristic matrix function

The known model of aj-unitary node with a givenj-inner characteristic matrix functionW is obtained from the known model of a unitary node with a given bi-inner characteristic matrix functionS, using the Potapov-Ginzburg transforms of the nodes and their characteristic functionsW andS. We show that some new properties of the L. de Branges reproducing kernal Hilbert spaces (W) which were discovered by H. Dym are characteristic properties of these spaces.     

Optimal time-variant systems and factorization of operators, I: Minimal and optimal systems

For a contractive block lower triangular operatorT optimal and star-optimal time-variant dissipative scattering realizations are introduced and their main properties are derived by state space considerations only. Two constructions of minimal and optimal realizations are given.     

Factorizations of and extensions to J-unitary rational matrix functions on the unit circle

This paper concerns two topics: (1) minimal factorizations in the class ofJ-unitary rational matrix functions on the unit circle and (2) completions of contractive rational matrix functions on the unit circle to two by two block unitary rational matrix functions which do not increase the McMillan degree. The results are given in terms of a special realization which does not require any additional properties at zero and at infinity. The unitary completion result may be viewed as a generalization of Darlington synthesis.     

Additive results for the Wg-Drazin inverse

In this paper we prove the formula for the expression (A+B)^d,W in terms of A,B,W,A^d,W,B^d,W, assuming some conditions for A,B and W. Here S^d,W denotes the generalized W-weighted Drazin inverse of a linear bounded operator S on a Banach space.     

Admissibility of Unbounded Operators and Wellposedness of Linear Systems in Banach Spaces

We study linear systems, described by operators A, B, C for which the state space X is a Banach space.We suppose that − A generates a bounded analytic semigroup and give conditions for admissibility of B and C corresponding to those in G. Weiss’ conjecture. The crucial assumptions on A are boundedness of an H^∞-calculus or suitable square function estimates, allowing to use techniques recently developed by N. Kalton and L. Weis. For observation spaces Y or control spaces U that are not Hilbert spaces we are led to a notion of admissibility extending previous considerations by C. Le Merdy. We also obtain a characterisation of wellposedness for the full system. We give several examples for admissible operators including point observation and point control. At the end we study a heat equation in X = L^p(Ω), 1 < p < ∞, with boundary observation and control and prove its wellposedness for several function spaces Y and U on the boundary ∂Ω.     

Extensions of bitriangular operators

We prove that every one-dimensional extension of a bitriangular operator has a cyclic commutant. We also prove that ifT is an extension of a bitriangular operator by an algebraic operator, then the weakly closed algebraW(T) generated byT has a separating vector.This work was partially supported by NSF Grant DMS-9401544.Participant, Workshop in Linear Analysis and Probability, Texas A&M University     

THE INTEGRAL COMPONENT OF AN ORDER BOUNDED TRANSFORMATION

Summary

Assume that T:L→M is a positive transformation from an arbitrary Riesz space L into a Dedekind complete Riesz space M, and let T=T_c + T_s be the decomposition of T into its integral component T_c and its singular component T_s.

In this note we give an elementary proof of the fact that for every o ? u ? L

T_c u = inf(s_nup Tu_n: o ? u_n + u),

under the conditionns stated above. This generalizes a recent result of C.D. Aliprantis [1].     

Scattering Systems with Several Evolutions and Multidimensional Input/State/Output Systems

The one-to-one correspondence between one-dimensional linear (stationary, causal) input/state/output systems and scattering systems with one evolution operator, in which the scattering function of the scattering system coincides with the transfer function of the linear system, is well understood, and has significant applications in H^∞ control theory. Here we consider this correspondence in the d-dimensional setting in which the transfer and scattering functions are defined on the polydisk. Unlike in the onedimensional case, the multidimensional state space realizations and the corresponding multi-evolution scattering systems are not necessarily equivalent, and the cases d = 2 and d > 2 differ substantially. A new proof of Andô’s dilation theorem for a pair of commuting contraction operators and a new statespace realization theorem for a matrix-valued inner function on the bidisk are obtained as corollaries of the analysis.     

The infinite-time admissibility of observation operators and operator Lyapunov equations

Let (A, –, C) be an abstract dynamical system withA being the generator of aC ₀-semigroup on a Hilbert spaceH, C:D(A)Y a linear operator,Y another Hilbert space. In this paper, some sufficient and necessary conditions are obtained for the observation operatorC to be infinite-time admissible. For a control system (A, B, –), due to duality argument, some sufficient and necessary conditions are also given for the control operatorB to be extended admissible. It is wellknown that observation operatorC is admissible if and only if the operator Lyapunov equation associated with the system has a nonnegative solution. In this paper, all nonnegative solutions to this equation are represented parametrically.This project is supported by the NNSF of China, and the Youth Science and Technique Foundation of Shanxi Province.     

The Weiss conjecture on admissibility of observation operators for contraction semigroups

We prove the conjecture of George Weiss for contraction semigroups on Hilbert spaces, giving a characterization of infinite-time admissible observation functionals for a contraction semigroup, namely that such a functionalC is infinite-time admissible if and only if there is anM>0 such that for alls in the open right half-plane. HereA denotes the infinitesimal generator of the semigroup. The result provides a simultaneous generalization of several celebrated results from the theory of Hardy spaces involving Carleson measures and Hankel operators.     

Nonlinearity inH ∞-control theory,causality in the commutant lifting theorem,and extension of intertwining operators

The problems studied in this note have been motivated by our work in generalizing linearH control theory to nonlinear systems. These ideas have led to a design procedure applicable to analytic nonlinear plants. Our technique is a generalization of the linearH theory. In contrast to previous work on this topic ([9], [10]), we now are able to explicitly incorporate a causality constraint into the theory. In fact, we show that it is possible to reduce a causal optimal design problem (for nonlinear systems) to a classical interpolation problem solvable by the commutant lifting theorem [8]. Here we present the complete operator theoretical background of our research together with a short control theoretical motivation.This work was supported in part by grants from the Research Fund of Indiana University, the National Science Foundation DMS-8811084 and ECS-9122106, by the Air Force Office of Scientific Research F49620-94-1-0098DEF, and by the Army Research Office DAAL03-91-G-0019 and DAAH04-93-G-0332     

A remark on mixedL 2/L ∞ bounds

This note shows that the Dym-Gohberg maximal entropy solution to the Nehari problem also satisfies a particularL ² bound. Using this bound we give a simple proof of the Kaftal-Larson-WeissL ²/L Nehari interpolation theorem.     

An eigenvalue problem for the numerical range of a bounded linear operator

LetX be a complex Lebesgue space with a unique duality mapJ fromX toX ^*, the conjugate space ofX. LetA be a bounded linear operator onX. In this paper we obtain a non-linear eigenvalue problem for (A)=sup{Re: W(A} whereW(A)={J(x)A(x)) : x=1}, under the assumption that (A) and the convex hull ofW(A) for some linear operatorsA onl _p , 2<p<.