# 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 issues Toeplitz operators of which the logo W is an m x m matrix functionality outlined on a disconnected curve r. The curve r is believed to be the union of s + 1 nonintersecting uncomplicated soft closed contours rOo r •. . . • rs which shape the certainly l orientated boundary of a finitely attached bounded area in t. Our major requirement at the image W is that on every one contour rj the functionality W is the restrict 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) should 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 dimension 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 capabilities j j Wj are normalized to I at infinity.

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B(>') -I}T (P rcp)( >. )+{A(>') -I}T (QrCP)(>'). )T. 1. I_A X )-1 B + m + + +. 5), respectively. 5! invertible, or equivalently, Then S if and only invertible. 2 factorization 2! )T=I +BT()'I_AT)-lCT . + m + + + 8et A x =A -B _ C _ and A! 5), respectively. ' equivalently, if Then 8 and only if I-PQ is invertible. 9). 3) gives an explicit formula for 8- 1. ). 5), respectively. In fact, in terms of [BGK 2] I-PQ is an indicator for the operator 8, as well as for the Toeplitz operator with symbol W.

Optimal Hankel norm model reductions and Wiener-Hopf Factorization I: The canonical case, SIAM J. , to appear. ; Gohberg, I. , Minimal Factorization of Matrix and Operator Functions, OTI Birkhiiuser, Basel, 1979. ; Gohberg, I. , The coupling method for solving integral equations, in Topics in Operator Theory Systems and Networks (ed. H. Dym and I. Gohberg), OT 12 Birkhauser, Basel, 1983, 39-73. A. , Factorization of transfer functions, SIAM l. Control and Opt. 18 (1980), 675-696. [CG] Clancey, K.

5! invertible, or equivalently, Then S if and only invertible. 2 factorization 2! )T=I +BT()'I_AT)-lCT . + m + + + 8et A x =A -B _ C _ and A! 5), respectively. ' equivalently, if Then 8 and only if I-PQ is invertible. 9). 3) gives an explicit formula for 8- 1. ). 5), respectively. In fact, in terms of [BGK 2] I-PQ is an indicator for the operator 8, as well as for the Toeplitz operator with symbol W. 2 an indicator for S pX IImPA: ImP-+ImP X, where P (resp. pX) IS IS given by the operator the spectral projection of A(resp.