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

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**Example text**

All C J~f+. 17 Let B be an operator on K:. Then B-f~+ E Afa if and only if B is a commuting expansion of A with respect to U. Proof. Assume B - f~+ E HI. Then B = (B - f2+) + f~+ E Afa + Af+ C AT+. Hence B commutes with U and BK:+ C K:+. The fact that (B - f~+)K:+ C K:+ E3 7/ implies that Pn(B - f~+)lT/= 0. 10. So P n B [ 7 / = A, and thus B is a commuting expansion of A with respect to U. To prove the converse, assume that B is a commuting expansion of A with respect to U. Then B EAf+ and P n B I T / = A.

12 Let {A; T, U} be the data for a commuting expansion problem, and assume that A + A* is strictly positive. 57) where G is an arbitrary operator in the set C(S+, S_, U). 54). 57) uniquely determine each other. The proof of this theorem uses the following lemmas. 13 Let F be an operator on ]~, and assume that F + F* is strictly positive. Then F is an invertible operator. Moreover, i f M is an invariant subspaee for F, then the operator F[ AA is an invertible operator on 3,l. In this case, F maps AJ one to one and onto At.

5) PnBI74 = X-" T k K E + * T *~ 9 k=O To see this let B()~) = ~ = o AkBk be the Taylor series expansion for/~. ~T*)-1 = v'r A k E +* T *k for A in the unit disc. Fix g E 7/. ~1-- 1) belongs to L~ and by matching coefficients we see that for n > 0 the coefficient of e ~n~ in the Fourier expansion of G is given by ~ = o B~E~-T*(k+~)" Using K* x-'~176 B 'kE *+ T *k Z-~k=0 it follows that PH=(E+)B()~) E+(I - s = E ;~K*T*~g = K * ( I - AT*)-lg. n-~O Recall that (~4g)(A) = E ~ ( I - AT*)-lg for IA[ < 1.

### 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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