Duality of the weak essential norm

Tylli, Hans-Olav

doi:10.1090/S0002-9939-00-05937-2

Duality of the weak essential norm
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Proc. Amer. Math. Soc. 129 (2001), 1437-1443 Request permission

Abstract:

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

A. G. Aksoy and L. Maligranda, Real interpolation and measure of weak noncompactness, Math. Nachr. 175 (1995), 5–12. MR 1355009, DOI 10.1002/mana.19951750102
Kari Astala, On measures of noncompactness and ideal variations in Banach spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 29 (1980), 42. MR 575533
Kari Astala and Hans-Olav Tylli, On the bounded compact approximation property and measures of noncompactness, J. Funct. Anal. 70 (1987), no. 2, 388–401. MR 874062, DOI 10.1016/0022-1236(87)90118-2
Kari Astala and Hans-Olav Tylli, Seminorms related to weak compactness and to Tauberian operators, Math. Proc. Cambridge Philos. Soc. 107 (1990), no. 2, 367–375. MR 1027789, DOI 10.1017/S0305004100068638
P. G. Casazza and H. Jarchow, Self-induced compactness in Banach spaces, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), no. 2, 355–362. MR 1386867, DOI 10.1017/S0308210500022770
Jesús M. F. Castillo and Manuel González, Three-space problems in Banach space theory, Lecture Notes in Mathematics, vol. 1667, Springer-Verlag, Berlin, 1997. MR 1482801, DOI 10.1007/BFb0112511
F. Cobos, A. Manzano and A. Martinez, Interpolation theory and measures related to operator ideals, Quart. J. Math. 50 (1999), 401-416.
F. Cobos and A. Martinez, Extreme estimates for interpolated operators by the real method, J. London Math. Soc. 60 (1999), 860–870.
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 341032, DOI 10.1090/S0002-9939-1973-0341032-5
Manuel González, Eero Saksman, and Hans-Olav Tylli, Representing non-weakly compact operators, Studia Math. 113 (1995), no. 3, 265–282. MR 1330211, DOI 10.4064/sm-113-3-265-282
Niels Grønbæk and George A. Willis, Approximate identities in Banach algebras of compact operators, Canad. Math. Bull. 36 (1993), no. 1, 45–53. MR 1205894, DOI 10.4153/CMB-1993-008-8
W. B. Johnson and J. Lindenstrauss, Correction to: “Some remarks on weakly compactly generated Banach spaces” [Israel J. Math. 17 (1974), 219–230; MR 54 #5808], Israel J. Math. 32 (1979), no. 4, 382–383. MR 571092, DOI 10.1007/BF02760467
Å. Lima, O. Nygaard and E. Oja, Isometric factorization of weakly compact operators and the approximation property, Israel J. Math. (to appear)
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces, Lecture Notes in Mathematics, Vol. 338, Springer-Verlag, Berlin-New York, 1973. MR 0415253
Michael J. Meyer, On a topological property of certain Calkin algebras, Bull. London Math. Soc. 24 (1992), no. 6, 591–598. MR 1183316, DOI 10.1112/blms/24.6.591
Michael J. Meyer, Lower bounds for norms on certain algebras, Illinois J. Math. 39 (1995), no. 4, 567–575. MR 1361520
A. N. Pličko, Some properties of a Johnson-Lindenstrauss space, Funktsional. Anal. i Prilozhen. 15 (1981), no. 2, 88–89 (Russian). MR 617482
O. I. Reĭnov, How bad can a Banach space with the approximation property be?, Mat. Zametki 33 (1983), no. 6, 833–846 (Russian). MR 709222
Hans-Olav Tylli, The essential norm of an operator is not self-dual, Israel J. Math. 91 (1995), no. 1-3, 93–110. MR 1348307, DOI 10.1007/BF02761641
P. Wojtaszczyk, Banach spaces for analysts, Cambridge Studies in Advanced Mathematics, vol. 25, Cambridge University Press, Cambridge, 1991. MR 1144277, DOI 10.1017/CBO9780511608735