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