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