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

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Since OG is a strict contraction, I + ~ G is invertible and its inverse is given by co (I + OG) - I = ~ ( O C ) n. n=O Since/C+ is closed and invariant under (~G) ~ for each n >__0, we conclude that (I + OG) -1 leaves/r invariant too. This completes the proof. 12. We split the proof into two parts. 57) is a commuting expansion of A and B + B* is strictly positive. The second part concerns the converse implication. PART 1. Let G E C(~+, g_, U) be given, and set 9 = R-1L. 15 we know that the operator I + aSG is invertible.

Therefore, P u i 2 + 1 7 / = A , and the proof is complete. 6 The parameterization of all solutions In this section we describe the set of all solutions to the positive real commuting expansion problem for the data set {A; T, U}. For this purpose we need the following set of operators o n ]~: C(E+, g_, U) = {G on ]C : G U = U G , G]C+ C ]C+ @ 7 / a n d IIG[I < 1}. 52) Since ]C+ -- A//+(C+) and ]C+@7/-- UA//+ (C_), an operator G belongs to the set C(E+, C_, U) if and only if G commutes with U and the (g+, g_)-symbol r of G belongs to H~~ g_), vanishes at zero, and its H~-norm I]r is strictly less than one.

30) PROOF. By our hypotheses R(A) and ~(~)--1 are analytic and uniformly bounded on [A[ < 1. AA+(E+). 4) that Rs = XXol/2E+ is in 7/. _ for all integers n > 0. s into K:_. However, because of stability,/C_ = Ad_(E_). This completes the proof. 4 A solution to the positive extension problem (2) Let iT, U; Q} be a data set for a positive extension problem. 2, the triple iT*, U*; Q} is also a data set for a positive extension problem. 20). Now let f~ be an operator on/C. Then ft commutes with U if and only if f~ commutes with U*.

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