On causality and linear maps

Linear operators
H
that take some space of signals
S
into itself, and the standard concepts of time invariance, invertibility, and causality, play a basic role in the study of systems. In this note
S
is taken to be a certain very general linear space of
n
-vector-valued functions defined on
[0, \infty )
. We prove and discuss a theorem which shows, in particular, that in a setting very often of interest, the inverse
H^{1}
of a linear invertible time-invariant causal
H
is in fact causal.