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A Class of Invariant Unitary Operators

✍ Scribed by Luis Báez-Duarte

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
117 KB
Volume
144
Category
Article
ISSN
0001-8708

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✦ Synopsis

Let H=L 2 ((0, ), dx), and K * f (x)= f (*x), for *>0, f # H. An invariant operator on H is one commuting with all the K * . A skew root is a self-adjoint, unitary operator on H satisfying T 2 =I, and TK * =K * *T, for all *>0. A generator g is an element of H such that the smallest, closed subspace containing [K * g] *>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. Nat. 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.

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