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Difference between revisions of "Evolution operator"

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A linear operator-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036700/e0367001.png" /> of two variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036700/e0367002.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036700/e0367003.png" /> that satisfies the properties 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036700/e0367004.png" />; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036700/e0367005.png" />; and 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036700/e0367006.png" />.
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A linear operator-function 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) $U(t,s) = U(s,t)^{-1}$.
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====Comments====
 
====Comments====
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In general, an evolution operator can be defined as a (not necessarily linear) operator-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036700/e0367007.png" /> satisfying 1) and 2). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036700/e0367008.png" /> are not subjected to restrictions, 3) is satisfied automatically. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036700/e0367009.png" /> belong to an infinite-dimensional space, the restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036700/e03670010.png" /> is a natural one, and the inverse need not exist at all.
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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. If t,s belong to an infinite-dimensional space, the restriction t \ge s is a natural one, and the inverse need not exist at all.
  
 
====References====
 
====References====
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<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>
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<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>
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Revision as of 18:39, 16 October 2017

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. If t,s belong to an infinite-dimensional space, 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=11865
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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