# Evolution operator

From Encyclopedia of Mathematics

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

[a1] | A. Pazy, "Semigroups of linear operators and applications to partial differential equations" , Springer (1983) pp. Chapt. 5 |

**How to Cite This Entry:**

Evolution operator.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Evolution_operator&oldid=42088

This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article