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
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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) |
Absolutely summing operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Absolutely_summing_operator&oldid=18591