The uniform continuity of certain translation semigroups

Johnson, Jimmie Lee

doi:10.1090/S0002-9939-1978-0512908-5

The uniform continuity of certain translation semigroups
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Proc. Amer. Math. Soc. 71 (1978), 197-203 Request permission

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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Self-adjoint Toeplitz operators