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A class of invariant unitary operators. (English) Zbl 0978.47025
Summary: Let \({\mathcal H}= L_2((0,\infty),dx)\), and \(K_\lambda f(x)= f(\lambda x)\), for \(\lambda>0\), \(f\in{\mathcal H}\). An invariant operator \({\mathcal H}\) is one commuting with all the \(K_\lambda\). A skew root is a selfadjoint, unitary operator on \({\mathcal H}\) satisfying \(T^2= I\), and \(TK_\lambda= K^*_\lambda T\), for all \(\lambda> 0\). A generator \(g\) is an element of \({\mathcal H}\) such that the smallest, closed subspace containing \(\{K_\lambda g\}_{\lambda> 0}\) is equal to \({\mathcal H}\). 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 \({\mathcal H}\) arising from a theorem due to A. Beurling [Proc. Nat. Acad. Sci. U.S.A. 41, 312-314 (1955; Zbl 0065.30303)] and B. Nyman [“On some groups and semigroups of translations”, Thesis, Uppsala (1950)] which shows the Riemann hypothesis is equivalent to a closure problem in Hilbert space.

47B38 Linear operators on function spaces (general)
47A15 Invariant subspaces of linear operators
Zbl 0065.30303
Full Text: DOI
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