Absolutely summing operator

A linear operator acting from a Banach space into a Banach space is called absolutely -summing () if there is a constant such that

whenever and . Here, denotes the value of the linear functional (the Banach dual of , cf. Adjoint space) at the element . The set of such operators, denoted by , becomes a Banach space under the norm , and is a Banach operator ideal. If , then .

The prototype of an absolutely -summing operator is the canonical mapping , where is a Borel measure on a compact Hausdorff space . In this case, .

The famous Grothendieck theorem says that all operators from into any Hilbert space are absolutely -summing.

Absolutely -summing operators are weakly compact but may fail to be compact (cf. also Compact operator). For a Hilbert space it turns out that is just the set of Hilbert–Schmidt operators (cf. Hilbert–Schmidt operator). Nuclear operators (cf. Nuclear operator) are absolutely -summing. Conversely, the product of absolutely -summing operators is nuclear, hence compact, if . This implies that the identity mapping of a Banach space is absolutely -summing if and only if (the Dvoretzky–Rogers theorem).

References

[a1]	J. Diestel, H. Jarchow, A. Tonge, "Absolutely summing operators" , Cambridge Univ. Press (1995)
[a2]	G.J.O. Jameson, "Summing and nuclear norms in Banach space theory" , Cambridge Univ. Press (1987)
[a3]	A. Pietsch, "Operator ideals" , North-Holland (1980)