Absolutely summing operator

A linear operator $ T $ acting from a Banach space $ X $ into a Banach space $ Y $ is called absolutely $ p $- summing ( $ 1 \leq p < \infty $) if there is a constant $ c \geq 0 $ such that

$$ \left ( \sum _ {k =1 } ^ { n } \left \| {Tx _ {k} } \right \| ^ {p} \right ) ^ {1/p } \leq $$

$$ \leq c \sup \left \{ {\left ( \sum _ {k = 1 } ^ { n } \left | {\left \langle {x _ {k} ,a } \right \rangle } \right | ^ {p} \right ) ^ {1/p } } : {a \in X ^ \prime , \left \| a \right \| \leq 1 } \right \} $$

whenever $ x _ {1} \dots x _ {n} \in X $ and $ n = 1,2, \dots $. Here, $ \langle {x _ {k} ,a } \rangle $ denotes the value of the linear functional $ a \in X ^ \prime $( the Banach dual of $ X $, cf. Adjoint space) at the element $ x _ {k} \in X $. The set of such operators, denoted by $ \Pi _ {p} ( X,Y ) $, becomes a Banach space under the norm $ \pi _ {p} ( T ) = \inf c $, and $ \Pi _ {p} = \cup _ {X,Y } \Pi _ {p} ( X,Y ) $ is a Banach operator ideal. If $ 1 \leq p < q < \infty $, then $ \Pi _ {p} \subset \Pi _ {q} $.

The prototype of an absolutely $ p $- summing operator is the canonical mapping $ { { \mathop{\rm Id} } } : {C ( K ) } \rightarrow {L _ {p} ( K, \mu ) } $, where $ \mu $ is a Borel measure on a compact Hausdorff space $ K $. In this case, $ \pi _ {p} ( { { \mathop{\rm Id} } } : {C ( K ) } \rightarrow {L _ {p} ( K, \mu ) } ) = \mu ( K ) ^ {1/p } $.

The famous Grothendieck theorem says that all operators from $ L _ {1} ( K, \mu ) $ into any Hilbert space are absolutely $ 1 $- summing.

Absolutely $ p $- summing operators are weakly compact but may fail to be compact (cf. also Compact operator). For a Hilbert space $ H $ it turns out that $ \Pi _ {p} ( H,H ) $ is just the set of Hilbert–Schmidt operators (cf. Hilbert–Schmidt operator). Nuclear operators (cf. Nuclear operator) are absolutely $ p $- summing. Conversely, the product of $ 2n $ absolutely $ p $- summing operators is nuclear, hence compact, if $ 2n \geq p $. This implies that the identity mapping of a Banach space $ X $ is absolutely $ p $- summing if and only if $ { \mathop{\rm dim} } ( X ) < \infty $( 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)