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Absolutely summing operator

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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)
How to Cite This Entry:
Absolutely summing operator. A. Pietsch (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Absolutely_summing_operator&oldid=18591
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098