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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.

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