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)