Conditions for the causality of nonlinear operators defined on a function space
This note considers nonlinear operators T with both range and domain subsets of the space of complex n-vector-valued functions of the real variable t for — co <t< oo. Conditions, in which energy-type quantities play a key role, are presented under which T is causal in the following sense: Let 33(T) denote the domain of T and let t0 = sup {<' | for all / t D(T), jit) = 0 for almost all t < t'}. Then T is causal if for an arbitrary 8 > t„, Tj = Tg a.e. on (t0, 5) whenever f and g belong to SD(7') and / = g a.e. on (t0 , S). 1. Notation. Let 3C„ denote the space of complex measurable n-vector-valued functions of the real variable t for — oo < t < co. The complex-conjugate transpose of an arbitrary / e 3C„ is written as /*. With g and h arbitrary elements of 3C„ , and x either an arbitrary real number or let (g, h; x) denote