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Proceedings of the American Mathematical Society

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The uniform continuity of certain translation semigroups

Author: Jimmie Lee Johnson
Journal: Proc. Amer. Math. Soc. 71 (1978), 197-203
MSC: Primary 47D05; Secondary 47A15
MathSciNet review: 0512908
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Abstract: Let $ {S_h}f(x) = f(x + h)$ for $ h \geqslant 0$, for $ f \in {L^2}({R^ + };K)$, where K is a separable Hilbert space. The translation semigroup $ {S_h}$ when restricted to an invariant subspace L is uniformly continuous if and only if $ {G_L}$ is an inner function and has an analytic continuation across an open arc of the unit circle at $ z = 1$. The operator-valued function $ {G_L}$ is associated with the invariant subspace L by Beurling's theorem.

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Keywords: Invariant subspace, semigroup of operators, Hardy spaces, Fourier transform, inner function, unilateral shift
Article copyright: © Copyright 1978 American Mathematical Society