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

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Since f~+ is a commuting expansion of A, the subspace ]C+ is invariant for ~+. 60), it follows that ]C+ is also an invariant subspace for B. We have already seen that B + B* is strictly positive. 57) is a solution of the positive real commuting expansion problem. PART 2. In this part we assume that B is a commuting expansion of A with respect to U and B + B* is strictly positive. 57). 61) G = -(I + L*AL)-IL*AR, where A = B - f~+. We claim that the operator I + L * A L is invertible, and thus, G is a well defined operator.

52) we see that ~ is an arbitrary function in H~176 satisfying I1~]1oo < 1. 12. This completes the proof. 4 Generalized Carath@odory interpolation This chapter deals with interpolation. 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.

In this section, we will review some classical results from the state space theory for input-output systems. For further results on state space methods see Chen [3], DeCarlo, [5], Kailath [16] and Rugh [17]. Let Z be an operator on 7-/ and B be an operator from b/ into 7/, while C maps 7-/into y and D maps b/into y. Then we say that {Z, B , C , D} is a state space realization of F if F is the function defined by F(A) = D + A C ( I - A Z ) - I B . Notice that the values of F are linear operators from b/into y and F is an analytic function in some neighborhood of the origin.

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