Metric Constrained Interpolation, Commutant Lifting and by C. Foias, A.E. Frezho, I. Gohberg, M.A. Kaashoek

By C. Foias, A.E. Frezho, I. Gohberg, M.A. Kaashoek

This publication provides a unified technique for fixing either desk bound and nonstationary interpolation difficulties, in finite or limitless dimensions, according to the commutant lifting theorem from operator conception and the kingdom area strategy from mathematical procedure thought. at first the authors deliberate a couple of papers treating nonstationary interpolation difficulties of Nevanlinna-Pick and Nehari style via lowering those nonstationary difficulties to desk bound ones for operator-valued features with operator arguments and utilizing classical commutant lifting concepts. This aid approach required us to check and additional boost the classical effects for the desk bound difficulties during this extra basic framework. right here the approach conception grew to become out to be very worthy for establishing the issues and for supplying traditional kingdom house formulation for describing the suggestions. during this approach our paintings concerned us in a wider application than unique deliberate. the ultimate result of our efforts are provided right here. The monetary aid in 1994 from the "NWO-stimulansprogramma" for the Thomas Stieltjes Institute for arithmetic within the Netherlands enabled us to begin the learn which bring about the current publication. We additionally gratefully recognize the aid from our domestic associations: Indiana college at Bloomington, Purdue college at West Lafayette, Tel-Aviv collage, and the Vrije Universiteit at Amsterdam. We warmly thank Dr. A.L. Sakhnovich for his rigorously examining of a big a part of the manuscript. eventually, Sharon clever ready very successfully and with nice care the troff dossier of this manuscript; we're thankful for her very good typing.

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Additional info for Metric Constrained Interpolation, Commutant Lifting and Systems

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In other words, for i = 1, ... ) - Bi-v,j] v=l extends to an analytic function onID. 24). We conclude that for i = 1, ... I - Zj)-V[Ci_v,j - Bi-v,j] extends to an analytic function on ID. 22) are fulfilled. 25) holds if and only if there exists a G in HOO(Y, V) satisfying (BG)(Z)left = Band IIGlioo ~ 'Y. Therefore the left tangential Herrnite-Fejer interpolation problem can be viewed as a special case of the standard left Nevanlinna-Pick problem. This completes the proof. 6. THE NEHARI EXTENSION PROBLEM Let K1 , K2 , K3 , ...

Are the Taylor coefficients of G. 5) converges in the operator norm. If the operators Bi,j are taken to be zero operators for i = 1, ... , rj - 1 and j = 1, ... , N, then the previous Hermite-Fejer problem will be called the left tangential Schur-Fejer interpolation problem. In particular, by choosing ZI =0 and N = 1 this contains the following tangential version of the classical block Schur interpolation problem considered at the end of Section 3. Given an operator B from f{l to X 1 and a set of operators Bi from y to X 1 for i = 0, 1, ...

2) TWO-SIDED NUDELMAN INTERPOLATlON Sec. 9 39 and the following coupling condition is satisfied = L . 3) j,k=O where Go, G 1 , G 1, ... are the Taylor coefficients of G. 3), then zr= i i ZiBGj+kCAk-BC and rA= j,k=O ziBGj+kcAk-13C. j,k=O Hence for the two-sided Nudelman interpolation problem to be solvable the operator r must satisfy the following Sylvester equation: - - zr - r A = BC - BC . 4) This sets the stage for the following existence result. 1. 4) holds. 6) is positive. 2. Let Q and {A *, C* ), respectively.

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