View source for Condensing operator

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" />.

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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. Kuratowski,   "Sur les espaces complets"  ''Fund. Math.'' , '''15'''  (1930)  pp. 301–309</TD></TR></table>