Bounded Hankel Operators

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3.3 Spectral Structure 113

and gf in L1 (S 1 ) (recall that the product of two functions in an L2 space is in L1 ; see [47, p. 66]) have Fourier coecients dierent from 0 at most in positions {1, 2, 3, . . .}. But gf f g, so gf and its conjugate each have nonzero Fourier coecients at most in positions {1, 2, 3, . . .}. Therefore gf is the constant function 0. Since f 0, the F. and M. Riesz theorem (Theorem ) implies that the set on which f vanishes has measure 0. Therefore g equals 0 almost everywhere. Since is not 0 there is a set of positive measure on which g vanishes. Hence g vanishes on a set of positive measure, and it is therefore 0 by the F. and M. Riesz...

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