K.-H. Förster

Nonnegative realizations of matrix transfer functions

Abstract Various aspects of the nonnegative, finite-dimensional realizability of time-invariant discrete linear systems are considered. A new proof of the basic result of the nonnegative realizability of a primitive (scalar-valued) transfer function with nonnegative impulse response function is given. An algorithm for establishing whether a scalar-valued transfer function with nonnegative impulse response has a nonnegative realization is presented. The main result characterizes the nonnegative realizability of a scalar-valued transfer function with the help of primitive transfer functions, and is extended to the general case of matrix-valued transfer functions. Then conditions for the existence of some special nonnegative realizations of transfer functions are presented, e.g., where the middle (main) matrix is irreducible, strictly positive or primitive.

LOCAL SPECTRAL RADII AND COLLATZ-WIELANDT NUMBERS OF MONIC OPERATOR POLYNOMIALS WITH NONNEGATIVE COEFFICIENTS

Abstract Operator polynomials L(λ) = λlI − λl−1Al−1 − … − λA1 − A0 are considered, where A0, …, Al−1 are nonnegative operators in a Banach space x with normal cone x+. For x ∈ x+ we define the local spectral radius rL(x) and the lower and upper Collatz-Wielandt numbers r L (x) and r L (x) , respectively, of x with respect to L. We characterize these quantities with the help of corresponding quantities with respect to the first companion operator belonging to L and the operator function S(λ) = Al−1 + λ−1Al−2 + … + λ−l+1A0. Many properties known in the linear case l = 1 have generalizations to the case l > 1; e.g., r L (x) ≤ r L (x) ≤ r L (x) is true for all x ∈ x+. From these local results we obtain results for the global spectral radius r(L), which were proved earlier under more restrictive conditions.

Spectral Properties of Operator Polynomials with Nonnegative Coefficients

We study properties of the polynomial Q(λ) = λmI − S(λ) where S(λ) = λlAl + ...+ λA1 + A0 and 1 ≤ m < l. The coefficients Al,...,A0 are in the positive cone of an ordered Banach algebra or are positive operators on a complex Banach lattice E and I is the identity. We study the properties of the spectral radius of S(λ) if λ is a nonnegative real number, and its connection with the existence of spectral divisors with nonnegative coefficients in the considered sense. We prove factorization results for nonnegative elements in an ordered decomposing Banach algebra with closed normal algebra cone and in the Wiener algebra. Earlier results on monic (nonnegative) operator polynomials are applied to the operator polynomial class studied here.

On nonnegative realizations of rational matrix functions and nonnegative input-output systems

We consider realizations (A, B, C) of rational l × κ matrix functions W, i.e. W(λ) = C(λ — A)-1 B if λ ∉ σ(A). Generalizing the scalar case we prove that W has a realization with nonnegative matrices iff for each of its minimal realizations (A 0, B 0, C 0) there exists an A 0-invariant polyhedral proper cone P 0 with B 0ℝκ + ⊂ P 0 and C 0 P 0 ⊂ ℝl +. For such functions W the McMillan degree is less than or equal to the size of nonnegative-minimal realizations, and the latter is less than or equal to the minimum of the number of extreme rays of polyhedral cones P 0 with the properties mentioned above. We give an example for which both these inequalities are strict.

Spectral properties of rational matrix functions with nonnegative realizations

Abstract If ( A,B,C ) is an (entrywise) nonnegative realization of a rational matrix function W (i.e. W ( λ ) = C ( λ − A ) −1 B for λ ∉ σ ( A )) vanishing at infinity, then r ( W ) := inf{ r ⩾ 0: W has no poles λ with r λ |} is a pole of W and r ( A ) := spectral radius of A is an eigenvalue of A . We prove that, if the realization is minimal-nonnegative, then 1. 1. r ( W ) = r ( A ), 2. 2. order of the pole r ( W ) of W = order of the pole r ( A ) of (· − A ) −1 . We characterize the order of these poles in the spirit of Rothblums index theorem, namely as the length of the longest chains of singular vertices in the reduced graph of A with a suitable new access relation, which incorporates B and C into the familiar access relation of A .

On spectra of expansion graphs and matrix polynomials

Abstract An expansion graph of a directed weighted graph G0 is obtained by replacing certain edges of G0 by disjoint chains. The adjacency matrix of the expansion graph is a partial linearization of a monic matrix polynomial. We prove results on common properties of a monic operator polynomial and its partial linearization. The graph G0 is connected if and only if each expansion graph of G0 is connected; in this case we compute the index of imprimitivity of the adjacency matrix of some special expansion graphs of G0.

Nonnegative unitary operators

Unitary operators in Hilbert space map an orthonormal basis onto another. In this paper we study those that map an orthonormal basis onto itself. We show that a sequence of cardinal numbers is a complete set of unitary invariants for such an operator. We obtain a characterization of these operators in terms of their spectral properties. We show how much simpler the structure is in finite-dimensional space, and also describe the structure of certain isometries in Hilbert space.

The index of triangular operator matrices

For any triangular operator matrix acting in a direct sum of complex Banach spaces, the order of a pole of the resolvent (i.e. the index) is determined as a function of the coefficients in the Laurent series for all the (resolvents of the) operators on the diagonal and of the operators below the diagonal. This result is then applied to the case of certain nonnegative operators in Banach lattices. We show how simply these results imply the Rothblum Index Theorem (1975) for nonnegative matrices. Finally, examples for calculating the index are presented.

Decomposable matrix polynomials

Abstract Decomposability of a monic matrix polynomial is defined as it was by Colojoara and Foias for a single matrix (operator). It is shown that decomposable polynomials have properties close to those of linear ones. They are characterized in the general case and also when they are products of linear factors.

Stability of semi-Fredholm properties in complex interpolation spaces

In this paper, we show that for interpolation morphisms S and the complex interpolation method the set of all θ ∈ (0, 1) such that S[θ] is a semi-Fredholm operator is open and the nullities, deficiencies and the indices of S[θ] are locally constant on this set.