R Available online at www.sciencedirect.com J. Math. Anal. Appl. 293 (2004) 71–78 www.elsevier.com/locate/jmaa A characterization of absolutely summing operators by means of McShane integrable functions V. Marraffa 1 Department of Mathematics, Via Archirafi 34, 90123 Palermo, Italy Received 4 November 2003 Submitted by J.A. Ball Abstract Absolutely summing operators between Banach spaces are characterized by means of McShane integrable functions.  2004 Elsevier Inc. All rights reserved. Keywords: Pettis integral; McShane integral; Absolutely summing operator 1. Introduction In [15] the McShane integral of functions taking values in a locally convex space is studied. In particular (see [15, Theorem 4]) Fréchet spaces for which Henstock lemma holds true for every McShane integrable function have been characterized. The proof of the lemma is based on the property that a nuclear operator is absolutely summing. In [4] Diestel characterized absolutely summing operators between Banach spaces by means of Pettis integrable strongly measurable functions. It is known that the class of Pettis integrable strongly measurable functions is strictly included in the class of McShane integrable functions. Diestel at the Symposium on Real Analysis and Measure Theory in Ischia (2002) observed that with the technique of [15, Theorem 4], it could be possible to characterize absolutely summing operators by means of McShane integrable functions. In this paper we continue the investigation of such operators and we prove the characterization conjectured by Diestel (Theorem 5). Applying a recent E-mail address: marraffa@math.unipa.it. 1 Supported by MURST of Italy. 0022-247X/$ – see front matter  2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2003.12.029