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 thought to be the union of s + 1 nonintersecting basic tender closed contours rOo r •. . . • rs which shape the definitely l orientated boundary of a finitely hooked up bounded area in t. Our major requirement at the image W is that on each 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 thought (see. e. g . • [1]. bankruptcy 2) the rational matrix functionality Wj (which differs from contour to contour) will 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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1 A(>'). ). )(QrCP)(>') (8 2cp)(>. )={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.

3). 8) to get * 1W(oo) c = W+(00)= W+(0) *- 1W(oo). 7) we see that W+(0) *-1 = (W+(0)-1) * = I + (WC_A=1_B:Q)Z*A:(-(A:)-lB_W-1+PC:) ball and Ran 30 while we have already observed that * * 1 * W(oo)=(I-B _A _ - 0 Jw. 1)) = _(A~)-IB_ + PO: w + (A~)-lpA:-I0: w * ** *1* ***1* - P(A _ -0 _ B JA _ - 0 _w - PO _ B _A _ - O_w = (A~)-I[_B_ + PA: -1 0 : wJ. Thus = w-B *A * - 1 0 * w - WO A - 1 Z*B - - - + WO A - 1 Z*P A * - 10 * w - - Now use that QZ *P = ZQP = -I+Z to. 3). 1. ) of a special form. 2. I-A)-1 B is ~ pxq rational matrix function of McMillan degree n such that all poles of K are in the open unit disk D.

1) uniquely and the projection of X onto X_ along X+ is called the separating projection for S. In case S is bounded, S is exponentially dichotomous if and only if the spectrum o(S) of S does not meet the imaginary axis and then the separating projection is just the Riesz projection corresponding to the part of o(S) lying in the right half plane. In general, the condition that S is exponentially diochotomous involves a more complicated spectral splitting of the (possibly connected) extended spectrum of S.