Representation of shift-invariant operators on \(L^ 2\) by \(H^{\infty}\) transfer functions: An elementary proof, a generalization to \(L^ p\), and a counterexample for \(L^{\infty}\). (English) Zbl 0724.93021
Summary: We give an elementary proof of the well-known fact that shift-invariant operators on \(L^ 2[0,\infty)\) are represented by transfer functions which are bounded and analytic on the right open half-plane. We prove a generalization to Banach space-valued \(L^ p\)-functions, where \(1\leq p<\infty\). We show that the result no longer holds for \(p=\infty\).
MSC:
| 93B28 | Operator-theoretic methods |
References:
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