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A [[Linear operator|linear operator]]  $  T $
acting from a [[Banach space|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|linear functional]]  $  a \in X  ^  \prime  $(
the Banach dual of  $  X $, 
cf. [[Adjoint space|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|Borel measure]] on a compact [[Hausdorff space|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|Hilbert space]] are absolutely  $  1 $-
summing.

Absolutely  $  p $-
summing operators are weakly compact but may fail to be compact (cf. also [[Compact operator|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|Hilbert–Schmidt operator]]). Nuclear operators (cf. [[Nuclear operator|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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Diestel,   H. Jarchow,   A. Tonge,   "Absolutely summing operators" , Cambridge Univ. Press  (1995)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.J.O. Jameson,   "Summing and nuclear norms in Banach space theory" , Cambridge Univ. Press  (1987)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Pietsch,   "Operator ideals" , North-Holland  (1980)</TD></TR></table>