The equivalence of causal and ordinary invertibility for integral convolution operators (Q1874770)

Let \(S\) be a cone in \(\mathbb{R}^n\). A linear operator \(T\) acting in a function space on \(\mathbb{R}^n\) is said to be causal if the value \((Tx)(t)\) at an arbitrary point \(t \in \mathbb{R}^n\) is completely determined by the values of \(x\) on the set \(t-S\). An operator \(T\) is said to be causally invertible if the inverse \(T^{-1}\) exists and is also causal. It is shown that for the integral convolution operator \[ (Gx)(t)=\int_S g(s) x(t-s) ds \] with \(g \in L^1(S)\), the ordinary invertibility of the operator \(I+G\) in \(L^p(S)\) coincides with the causal invertibility for \(n\geq 2\).