Wiener--Hopf operators on spaces of functions on \({\mathbb{R}}^{+}\) with values in a Hilbert space (Q2476169)

Let \(E\) be a Banach space of functions on \(\mathbb{R}^+\) such that \(E\subset L^1_{\mathrm{loc}}(\mathbb{R}^+)\). For \(a\geq 0\), define the operator \(S_a: E\to L^1_{\mathrm{loc}}(\mathbb{R}^+)\) by the formula \((S_a f)(x)=f(x-a)\) for almost every \(x\in[a,\infty)\) and \((S_a f)(x)=0\) for \(x\in[0, a)\). For \(a\geq0\), define \(S_{-a}: E\to L^1_{\mathrm{loc}}(\mathbb{R}^+)\) by the formula \((S_{-a}f)(x)=f(x+a)\) for almost every \(x\in\mathbb{R}^+\). Suppose that \(S_aE\subset E\) and \(S_{-a}E\subset E\) for all \(a\in\mathbb{R}^+\). The Wiener--Hopf operators on \(E\) are the bounded operators \(T: E\to E\) satisfying \(S_{-a}TS_a=T\) for all \(a\in\mathbb{R}^+\). In this paper, the author obtains a representation theorem for Wiener--Hopf operators on a large class of functions on \(\mathbb{R}^+\) with values in a separable Hilbert space.