By Robert Hermann
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Example text
Set G G- 1 = [R(w),Alf - [R(z),Alf = (z - w)([R(w), AlR(z) + R(w)[R(z),AJ)f for all f E V(Hm). This implies [R(w), Al(I + (w - z)R(z))f = (I + (z - w)R(w))[R(z), Alf, f E V(H m ), which leads to [R(z), AlJ = G[R(w), AlGf, We show by induction that for all integers k Indeed, suppose (9) holds for a fixed k ~ ° f E V(Hm). ~ 0, (the case k = ° is trivial). (8) D. Cichon, J. H. Szafraniec 24 Then, applying (8) to the operator (adR(w))k(A) in place of A and (9), we get (adR(z))k+l(A)f = [R(z), (adR(z))k(A)lJ = [R(z),Ck(adR(w))k(A)CklJ = Ck[R(z), (adR(w))k(A)]C k f Ck+l[R(w), (adR(w))k(A)]C k+l f = Ck+l(ad R(w))k+l (A)C k+l f, f E V(Hffi).
3) where no = n(Ao), aj(A)(laj(A)1 = l;j = 1, ... , no) are continuous functions on A, aj(A) -I ak(A)(A E A; j -I k, j, k = 1, ... (z) -I 0 for 1Z 1= 1 and A E A. (V), from ~(V). (V)(A E A). 4) j=1 According to the restrictions V, the operators V* - aj(A)I (j = 1, ... ,no) are oneto-one, and their corresponding inverses Rj(A) = (V* - aj(A)I)-1 (j = 1, ... ,no) are closed and unbounded operators in the space Ji. 1 Rj(A) (A E A). Also, it is easy to see that the operator A>. (V) is invertible on the left, and let A~-1) (E B(Ji)) denote its left inverse.
Feldman [18], see also S. Prossdorf [24]. According to the theory developed in [18], Wiener-Hopf type operators can be regarded in a certain sense as functions of one-sided invertible operators. As our case is that of Hilbert's space, the considered operators will be presented as functions of an isometric operator. Moreover, these functions (so-called symbols of the corresponding operators) will be assumed to be continuous on the unit circle of the complex plane. In order to specify the notion connected with the foregoing discussion, let 1i be an arbitrary Hilbert space and consider on it an isometric operator denoted by V.