Completely-continuous operator

Completely-Continuous Operator

A bounded linear operator $f$, acting from a Banach space $X$ into another space $Y$, that transforms weakly-convergent sequences in $X$ to norm-convergent sequences in $Y$. Equivalently, an operator $f$ is completely-continuous if it maps every relatively weakly compact subset of $X$ into a relatively compact subset of $Y$. It is easy to see that every compact operator is completely continuous, however the converse is false. For example, recall that the Banach space $X=l_1$ 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. The term "completely-continuous operator" originally meant what we now call "compact operator", which has sometimes resulted in confusion.

It can be assumed that the space $X$ is separable (for $Y$ 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)
[4]	R. E. Megginson, "An Introduction to Banach Space Theory" , Springer (1998) pp. 336-339
[5]	A. Pietsch, "History of Banach Spaces and Linear Operators" , Birkhauser (2007) pp. 49-50

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)