A class of invariant unitary operators

Let H =L-2((0, infinity), dx), and K(lambda)f(x) = f(lambda x), for lambda>0, f is an element of H. An invariant operator, on H is one commuting with all the K-lambda. A skew root is a self-adjoint, unitary operator on H satisfying T-2 = I, and TKlambda = K-lambda*T, for all lambda > 0. A generator g is an element of H such that the smallest, closed subspace containing {K-lambda g}(lambda>0) is equal to H. We show that for any skew root T and any real-valued generator g there is a unique, invariant, unitary operator W satisfying Wg = Tg. It turns out that W-1 = TWT. This construction is related to an approximation problem in H arising from a theorem due to A. Beurling (1955, Proc. Nar. Acad. Sci. U.S.A. 41, 312-314) and B. Nyman (1950, "On Some Groups and Semigroups of Translations," Thesis, Uppsala) which shows the Riemann hypothesis is equivalent to a closure problem in Hilbert space. (C) 1999 Academic Press.