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By Frazho A.E., Kaashoek M.A.

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Extra resources for A band method approach to a positive expansion problem in a unitary dilation setting

Example text

8), we see that (FF)(Z)left = WTF[H and F = W[ H where H is embedded in the first component of g~(H). From this it readily follows that (FF)(Z)left = F if and only if WTF = W. 10) if and only if F is a function in H~(/4,/4) satisfying the following two conditions WTF = W and L F + L*F > 51 > 0. 17) Hence F is a solution to our generalized Carath6odory interpolation problem with data {Z, F, F} if and only if F is a function in H ~ ( U , Lt) such that W T F = W and LF + L*F >_ 5 I for some 6 > 0.

In the first section it is shown that the problem of finding commuting expansions can be rephrased as a tangential interpolation problem for operator-valued H ~176 functions on the open unit disc with interpolation at operator points. In the second section we use our main theorems to solve explicitly a generalized Carath~odory interpolation problem, that is, a Nevanlinna-Pick interpolation problem with the usual norm constraint being replaced by a positivity condition. 1 Commuting expansions and interpolation We need the notion of point evaluation for an analytic operator-valued function.

1 Therefore Do = D1. 41) along with J ' W * = W ' Z 1 , C o W * = - C 1 and Do = D1, we obtain 5+(A) = Do - ACo(I - A J * ) - I T * X X o 1 = D1 - ACo(I - A J * ) - I T * W * A - 1 F X o I = 91 - ACo(I - ~J*)-IW*Z*A-lrxo = DI + ACI(I - ~Z1)-lZ*A-lrx; I = D1 - ) , C o W * U - ~ Z 1 ) - I Z * A - l r X o 1 ~. 24). 10 completes the proof. To obtain a description of all solutions to our generalized Carathdodory interpolation problem we use the following result. 4 Let { Z on X, F} be a stable finite t i m e controllable pair.

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A band method approach to a positive expansion problem in a unitary dilation setting by Frazho A.E., Kaashoek M.A.

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