Evolution operator

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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) $U(t,s) = U(s,t)^{-1}$.


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.


[a1] A. Pazy, "Semigroups of linear operators and applications to partial differential equations" , Springer (1983) pp. Chapt. 5
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Evolution operator. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article