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Difference between revisions of "Condensing operator"

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An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024450/c0244501.png" />, generally non-linear, defined on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024450/c0244502.png" /> of all subsets of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024450/c0244503.png" /> in a normed vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024450/c0244504.png" />, with values in a normed vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024450/c0244505.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024450/c0244506.png" /> — the measure of non-compactness of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024450/c0244507.png" /> — is less than the measure of non-compactness <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024450/c0244508.png" /> for any non-compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024450/c0244509.png" />. Here, the measures of non-compactness may be the same in both cases or different. For example, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024450/c02445010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024450/c02445011.png" /> one may take the Kuratowski measure of non-compactness: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024450/c02445012.png" />.
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An operator  $  U $,  
 +
generally non-linear, defined on the set $  \mathfrak M $
 +
of all subsets of a set $  M $
 +
in a normed vector space $  X $,  
 +
with values in a normed vector space $  Y $,  
 +
such that $  \psi _ {Y} [ U ( A) ] $—  
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the measure of non-compactness of the set $  U ( A) \subset  Y $—  
 +
is less than the measure of non-compactness $  \psi _ {X} ( A) $
 +
for any non-compact set $  A \in \mathfrak M $.  
 +
Here, the measures of non-compactness may be the same in both cases or different. For example, as $  \psi _ {X} $
 +
and $  \psi _ {Y} $
 +
one may take the Kuratowski measure of non-compactness: $  \alpha ( A) = \inf \{ d > 0,  A  \textrm{ may  be  decomposed  into  finitely  many  subsets  of  diameter  less  than  }  d \} $.
  
 
For a continuous condensing operator many constructions and facts of the theory of completely-continuous operators can be carried over, for instance, the rotation of contracting vector fields, the fixed-point principle of contraction operators, etc.
 
For a continuous condensing operator many constructions and facts of the theory of completely-continuous operators can be carried over, for instance, the rotation of contracting vector fields, the fixed-point principle of contraction operators, etc.

Latest revision as of 17:46, 4 June 2020


An operator $ U $, generally non-linear, defined on the set $ \mathfrak M $ of all subsets of a set $ M $ in a normed vector space $ X $, with values in a normed vector space $ Y $, such that $ \psi _ {Y} [ U ( A) ] $— the measure of non-compactness of the set $ U ( A) \subset Y $— is less than the measure of non-compactness $ \psi _ {X} ( A) $ for any non-compact set $ A \in \mathfrak M $. Here, the measures of non-compactness may be the same in both cases or different. For example, as $ \psi _ {X} $ and $ \psi _ {Y} $ one may take the Kuratowski measure of non-compactness: $ \alpha ( A) = \inf \{ d > 0, A \textrm{ may be decomposed into finitely many subsets of diameter less than } d \} $.

For a continuous condensing operator many constructions and facts of the theory of completely-continuous operators can be carried over, for instance, the rotation of contracting vector fields, the fixed-point principle of contraction operators, etc.

References

[1] B.N. Sadovskii, "Limit-compact and condensing operators" Russian Math. Surveys , 27 (1972) pp. 85–155 Uspekhi Mat. Nauk , 27 : 1 (1972) pp. 81–146
[2] C. Kuratowski, "Sur les espaces complets" Fund. Math. , 15 (1930) pp. 301–309
How to Cite This Entry:
Condensing operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Condensing_operator&oldid=46439
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article