Weakly compact operators into sequence spaces: A counterexample

Ylinen, Kari

doi:10.1090/S0002-9939-04-07771-8

Weakly compact operators into sequence spaces: A counterexample
HTML articles powered by AMS MathViewer

by Kari Ylinen

Proc. Amer. Math. Soc. 133 (2005), 1423-1425

DOI: https://doi.org/10.1090/S0002-9939-04-07771-8

Published electronically: November 19, 2004

PDF | Request permission

Erratum: Proc. Amer. Math. Soc. 134 (2006), 311-311.

Abstract:

In a recent paper Gutiérrez and Villanueva have used, without giving a detailed proof, an analogue of a well-known result of Ryan characterizing the weakly compact operators from a Banach space $E$ into the space $c_0(X)$ of null sequences in a Banach space $X$. In this note a counterexample is given showing that in the statement of Gutiérrez and Villanueva an additional condition is needed.

References

Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
Joaquín M. Gutiérrez and Ignacio Villanueva, Extensions of multilinear operators and Banach space properties, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), no. 3, 549–566. MR 1983686, DOI 10.1017/S0308210500002535
A. Olubummo, Finitely additive measures on the non-negative integers, Math. Scand. 24 (1969), 186–194 (1970). MR 274642, DOI 10.7146/math.scand.a-10929
Raymond A. Ryan, Dunford-Pettis properties, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 5, 373–379 (English, with Russian summary). MR 557405
J. D. Maitland Wright and Kari Ylinen, Multilinear maps on products of operator algebras, J. Math. Anal. Appl. 292 (2004), no. 2, 558–570. MR 2047631, DOI 10.1016/j.jmaa.2003.12.024