Completely-continuous operator

Completely-Continuous Operator

A bounded linear operator , acting from a Banach space into another space , that transforms weakly-convergent sequences in to norm-convergent sequences in . Equivalently, an operator is completely-continuous if it maps every relatively weakly compact subset of into a relatively compact subset of . It is easy to see that every compact operator is completely continuous, however the converse is false. For example, recall that the Banach space X = l₁ has the Schur Property, that is weak sequential and norm sequential convergence coincide. It follows that the identity operator from X to X is completely-continuous, but it is not compact since X is infinite-dimensional. If X is reflexive, then every completely-continuous operator is compact, so the two classes of operators do coincide in that case. In the past, the term "completely-continuous operator" was often used to mean compact operator which has sometimes resulted in confusion.

It can be assumed that the space is separable (for this is not a necessary condition; however, the image of a completely-continuous operator is always separable).

The class of compact operators is the most important class of the set of completely-continuous operators (cf. Compact operator).

References

[1]	D. Hilbert, "Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen" , Chelsea, reprint (1953)
[2]	F. Riesz, "Sur les opérations fonctionelles linéaires" C.R. Acad. Sci. Paris Sér. I Math. , 149 (1909) pp. 974–977
[3]	S.S. Banach, "Théorie des opérations linéaires" , Hafner (1932)

References

[a1]	N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958)
[a2]	A.E. Taylor, D.C. Lay, "Introduction to functional analysis" , Wiley (1980)