Acta Math. Hungar., 129 (3) (2010), 227–244 DOI: 10.1007/s10474-010-0025-7 First published online November 3, 2010 GENERALIZED MEASURES OF NONCOMPACTNESS OF SETS AND OPERATORS IN BANACH SPACES E. B. DA SILVA 1 and D. L. FERNANDEZ 2 1 Universidade Estadual de Maring´ a – UEM, Departamento de Matem´ atica – Av, Colombo 5790, Maring´ a, PR, Brazil, 87020-900 e-mail: ebsilva@wnet.com.br 2 Universidade Estadual de Maring´ a – Unicamp, Departamento de Matem´ atica, Caixa Postal, 6065 Campinas, SP, Brazil, 13083-859 e-mail: dicesar@ime.unicamp.br (Received July 8, 2009; revised April 6, 2010; accepted April 12, 2010) Abstract. New measures of noncompactness for bounded sets and linear op- erators, in the setting of abstract measures and generalized limits, are constructed. A quantitative version of a classical criterion for compactness of bounded sets in Banach spaces by R. S. Phillips is provided. Properties of those measures are established and it is shown that they are equivalent to the classical measures of noncompactness. Applications to summable families of Banach spaces, interpola- tions of operators and some consequences are also given. 1. Introduction The notion of measure of noncompactness was introduced by K. Kura- towski. The Kuratowski measure, as well as a variant of it, called by some authors as Haudorff measure, have a very important role in functional analy- sis and they are applied to the theories of differential and integral equations as well as to the operator theory. This notion afterwards got an abstract setting in which the Kuratowski and Hausdorff measures are only examples, albeit the most important. For instance, in [1], [2], [3] and [8] general defini- tions of measures of noncompactness and their connections with fixed point theory, condensing operators and geometric properties of Banach spaces are presented. In this work we are considering an axiomatic definition of measure of noncompactness which is a variant of the definition in [3], being friendly and useful in applications. Through the concepts of generalized sequence Key words and phrases: measure of noncompactness, Banach space, partially ordered set, compact operator, summable family, interpolation space. 2000 Mathematics Subject Classification: 46B45, 46B50, 47H09, 46B70. 0236-5294/$ 20.00 c  2010 Akad´ emiai Kiad´o, Budapest, Hungary