Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Duality of the weak essential norm

Author: Hans-Olav Tylli
Journal: Proc. Amer. Math. Soc. 129 (2001), 1437-1443
MSC (2000): Primary 47A30, 46B20, 46B28
Published electronically: October 24, 2000
MathSciNet review: 1814170
Full-text PDF

Abstract | References | Similar Articles | Additional Information


It is established by an example that the natural quotient norms $S \mapsto\mathrm{dist}(S,W(E,F))$ and $S \mapsto\mathrm{dist}(S^{*},W(F^{*},E^{*}))$ are not comparable in general. Hence there is no uniform quantitative version of Gantmacher's duality theorem for weakly compact operators in terms of the preceding weak essential norm. Above $W(E,F)$ stands for the class of weakly compact operators $E\to F$, where $E$ and $F$ are Banach spaces. The counterexample is based on a renorming construction related to weakly compact approximation properties that is applied to the Johnson-Lindenstrauss space $JL$.

References [Enhancements On Off] (What's this?)

  • [AM] A.G. Aksoy and L. Maligranda, Real interpolation and measure of weak noncompactness, Math. Nachr. 175 (1995), 5-12. MR 96m:46138
  • [A] K. Astala, On measures of noncompactness and ideal variations in Banach spaces, Ann. Acad. Sci. Fenn. Ser. A I. Math. Dissertationes 29 (1980), 1 - 42. MR 83a:46027
  • [AT1] K. Astala and H.-O. Tylli, On the bounded compact approximation property and measures of noncompactness, J. Funct. Anal. 70 (1987), 388-401. MR 88c:47020
  • [AT2] K. Astala and H.-O. Tylli, Seminorms related to weak compactness and to Tauberian operators, Math. Proc. Cambridge Phil. Soc. 107 (1990), 367-375. MR 91b:47016
  • [CJ] P.G. Casazza and H. Jarchow, Self-induced compactness in Banach spaces, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), 355-362. MR 97a:46010
  • [CG] J.M.F. Castillo and M. González, Three-space problems in Banach space theory, Lecture Notes in Mathematics 1667 (Springer-Verlag, 1997). MR 99a:46034
  • [CMM] F. Cobos, A. Manzano and A. Martinez, Interpolation theory and measures related to operator ideals, Quart. J. Math. 50 (1999), 401-416. CMP 2000:05
  • [CM] F. Cobos and A. Martinez, Extreme estimates for interpolated operators by the real method, J. London Math. Soc. 60 (1999), 860-870. CMP 2000:11
  • [FJ] T. Figiel and W.B. Johnson, The approximation property does not imply the bounded approximation property, Proc. Amer. Math. Soc. 41 (1973), 197-200. MR 49:5782
  • [GST] M. González, E. Saksman and H.-O. Tylli, Representing non-weakly compact operators, Studia Math. 113 (1995), 265-282. MR 96i:47078
  • [GW] N. Grønbæk and G.A. Willis, Approximate identities in Banach algebras of compact operators, Canad. Math. Bull. 36 (1993), 45-53. MR 94b:46076
  • [JL] W.B. Johnson and J. Lindenstrauss, Some remarks on weakly compactly generated Banach spaces, Israel J. Math. 17 (1974), 219-230. MR 81g:46015
  • [LNO] Å. Lima, O. Nygaard and E. Oja, Isometric factorization of weakly compact operators and the approximation property, Israel J. Math. (to appear)
  • [LT] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, Lecture Notes in Mathematics 338 (Springer-Verlag, 1973). MR 54:3344
  • [M1] M.J. Meyer, On a topological property of certain Calkin algebras, Bull. London Math. Soc. 24 (1992), 591-598. MR 93i:46081
  • [M2] M.J. Meyer, Lower bounds for norms on certain algebras, Illinois J. Math. 39 (1995), 567-575. MR 97g:46063
  • [P] A.N. Plichko, Some properties of Johnson-Lindenstrauss space, Funct. Anal. Appl. 15 (1981), 149-150. MR 82h:46021
  • [R] O. Reinov, How bad can a Banach space with the approximation property be? Math. Notes 33 (1983), 427-434. MR 85m:46018
  • [T] H.-O. Tylli, The essential norm of an operator is not self-dual, Israel J. Math. 91 (1995), 93-110. MR 96f:47017
  • [W] P. Wojtaszczyk, Banach spaces for analysts, Cambridge Stud. Adv. Math. 25 (Cambridge University Press, 1991). MR 93d:46001

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47A30, 46B20, 46B28

Retrieve articles in all journals with MSC (2000): 47A30, 46B20, 46B28

Additional Information

Hans-Olav Tylli
Affiliation: Department of Mathematics, University of Helsinki, P. O. Box 4 (Yliopistonkatu 5), FIN-00014 University of Helsinki, Finland

Received by editor(s): August 17, 1999
Published electronically: October 24, 2000
Communicated by: Dale Alspach
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society