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3(a)j the differences are only in the details. 2(b). 2), and assume that conditions (CM1), (CM2) and (CM3b) are satisfied. We have to prove that all strictly contractive extensions are given by Fe(h), where h is an arbitrary element of N1 such that q-h*ph >n O. We split the proof into four parts. Part (Q). 20) 8i1N1 C pN1 , 8i2N1 C qN+. 3). 21) hold. Part ({3). Let h be an element of N1 such that q - h*ph is positive definite in n. In this part we show that f := Fe(h) is well-defined and f is a strictly contractive extension of k.

Thus kEN if I - k E Nl and R-positive real part extensions and strictly contractive extensions of k are elements of N. Therefore, the role of the algebra N is minor, and in what follows we may as well take N=R. Our aim is to derive linear fractional representations of all R-positive real part extensions of k and of all strictly contractive extensions of k. For this purpose we need an additional connection between N+ and R, which is expressed by the following axiom. AXIOM (A). If g E N+ and IIglln < 1, then (e - g)-l E N+.

Furthermore, vP-lV*Yl ( -(c+c*)v v = -VXl 0) u E vNl C N+, and thus x+ E N+. Finally, we have (Xl) X+ =( -vp Jc*+c*)Vp-lV*Y~l V Yl + Y+ + vp * V Yl ) = (Yl), Y+ J. A. Ball, I. Gohberg and M. A. 19). 18) holds, and (CM1) is proved. 3(b). 3(a)j the differences are only in the details. 2(b). 2), and assume that conditions (CM1), (CM2) and (CM3b) are satisfied. We have to prove that all strictly contractive extensions are given by Fe(h), where h is an arbitrary element of N1 such that q-h*ph >n O. We split the proof into four parts.

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