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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) ] $— 
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====
<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>