Positivity (2014) 18:29–39
Remarks on multiple summing operators
Received: 11 April 2012 / Accepted: 21 January 2013 / Published online: 9 February 2013
© Springer Basel 2013
Abstract We give the necessary and sufﬁcient conditions for a multilinear bounded
operator on C(
) × X
to be multiple 1-summing.
Based on this result we prove an inclusion result for multiple summing operators and
an unexpected composition result of Grothendieck type for bilinear operators.
Keywords p-summing operators · Multiple p-summing operators ·
Nuclear operators · Banach spaces of continuous functions
Mathematics Subject Classiﬁcation (2010) Primary 47H60 ; Secondary 47B10 ·
47L20 · 46G10
1 Introduction and notation
The notion of absolutely summing operators, called also 1-summing, which was ﬁrst
introduced by Grothendieck under the name ”semi-integrale á droite”, is now well
recognized and appreciated. We just mention the famous monographs of Defant and
Floret , Diestel, Jarchow and Tonge  and Pietsch , where the absolutely
summing operators play a fundamental role.
In the linear case, 1-summing operators deﬁned on C() or on C(, X) have
some very beautiful characterizations, see the famous monograph of Diestel and Uhl
[5, Chapter V], or Swartz’s paper .
A natural question is how can we extend the characterization of 1-summing linear
operators given in [5, Chapter V] to the multilinear case. In this paper we present some
D. Popa (
Department of Mathematics, Ovidius University of Constanta,
Bd. Mamaia 124, 900527 Constanta, Romania