Generalized Unitary Operators

1. Let C be the complex field and T be the unit circle { \ £ C : | \ | = 1} . For a non-negative integer m or for m = oo, let OCT) be the space of all m-times continuously differentiable functions on T. (Here we consider T as a C°°-manifold in the natural way. Thus, any / £ C m ( r ) can be identified with an m-times continuously differentiable periodic function f (6) of a real variable 0 with period 2TT.) C(T) is an algebra as well as a Banach space if m is finite, a Fréchet space if m = oo, with the usual sup-norms for derivatives. We shall say tha t a mapping y of T into C is a C-curve if y can be extended onto a neighborhood V of V (the extended map will also be denoted by 7) in such a way that it is one-to-one on V and y and 7"* are both w-times continuously differentiable (as functions in two variables) on Vand y(V) respectively. Let £ be a Hausdorff locally convex space over C such that the space <£(£) of all continuous linear operators on E endowed with the bounded convergence topology is quasi-complete.