Summary: Results are given that establish, for the first time, necessary and sufficient conditions for the existence of impulse responses and kernel representations for linear not-necessarily-time-invariant systems described by input-output operator equations.
These results concern systems whose inputs and outputs are real-valued functions on the real line \({\mathbb{R}}\), the half-line \([0,\infty)\) or \({\mathbb{R}}^ m\). They deal with causal as well as non-causal maps and considerably extend related previous results which concern causal maps defined on functions on the half-line.