Download e-book for iPad: Constructive Methods of Wiener-Hopf Factorization by I. Gohberg, M. A. Kaashoek

By I. Gohberg, M. A. Kaashoek

The major a part of this paper matters Toeplitz operators of which the logo W is an m x m matrix functionality outlined on a disconnected curve r. The curve r is thought to be the union of s + 1 nonintersecting uncomplicated tender closed contours rOo r •. . . • rs which shape the absolutely l orientated boundary of a finitely attached bounded area in t. Our major requirement at the image W is that on each one contour rj the functionality W is the limit of a rational matrix functionality Wj which doesn't have poles and zeros on rj and at infinity. utilizing the belief theorem from process idea (see. e. g . • [1]. bankruptcy 2) the rational matrix functionality Wj (which differs from contour to contour) could be written within the shape 1 (0. 1) W . (A) = I + C. (A - A. f B. A E r· J J J J J the place Aj is a sq. matrix of measurement nj x n• say. B and C are j j j matrices of sizes n. x m and m x n . • respectively. and the matrices A. J x J J and Aj = Aj - BjC haven't any eigenvalues on r . (In (0. 1) the services j j Wj are normalized to I at infinity.

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We call e exponential (1) ~ 00 = 00 be a (A,B',C) a realization triple ~ if the following conditions are satisfied: -iA is exponentially dichotomous of exponential type 00, ( 2) V(C) ;J (3) m there exists a linear operator Ae: X + L 1 (R) such that 7 nx! ;-iA)x, X € V( A) • Note that B, being a linear operator from ~m into X, is automatically bounded. 3) Lm l ,00 , 00 (R), where (R) Taking into account (ii) and the fact that V(A) one sees that Ae is determined uniquely. 3) is a linear subspace of L7(R).

1 (interchanging + and -), where W +(00) = W joo) I . m One easily sees that XjA):= 36 Ball and Ran Y + (oo)W jA) the factor for a right spectral factorization for W(A). 1. 1 to a function Y +(A) having a special form. 2. = C(AI-A)-l B is ~ stable ration·al pxq matrix function of Mc:Millan degree n. Thus we may assume that the spectrum of the nxn matrix A is in the open left half plane {ReA

In the present paper we treat mxm matrix functions ~Itl mxm k such that k belongs to the class e L1 (R) for some A ~ < O. 1), but now the operators A,B and C have the following properties. 2) -iCe -i tApx, t > 0, t < 0, maps the domain of A into the space of all functions in e~ltIL~(R) with a derivative in L~(R), and extends to a bounded linear operator defined on X with values in e~ltIL~(R). 1). To use the method of factorization and inversion referred to above for the class of functions considered in the present paper, the main difficulty is to prove the following result.

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