View source for Evolution operator

A linear operator-function $U(t,s)$ of two variables $t$ and $s$ that satisfies the properties 1) $U(s,s) = I$; 2) $U(t,x)U(x,s) = U(t,s)$; and 3) $U(t,s) = U(s,t)^{-1}$.


====Comments====
In general, an evolution operator can be defined as a (not necessarily linear) operator-function $U(t,s)$ satisfying 1) and 2). If $t,s$ are not subjected to restrictions, 3) is satisfied automatically. Under some circumstances, the restriction $t \ge s$ is a natural one, and the inverse need not exist at all.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Pazy,   "Semigroups of linear operators and applications to partial differential equations" , Springer  (1983)  pp. Chapt. 5</TD></TR>
</table>

{{TEX|done}}