# Condensing operator

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