Abstract
We study two semigroups\(\mathcal{U}^{up} _ + ,\mathcal{R}^{up} _ + \) of operators between Banach spaces, related with the finite representability ofc 0 and ℓ1. We show that these semigroups are open, have nice duality properties and can be characterized in terms of compact perturbations, and in terms of the properties of their ultrapowers. We obtain analogous results for their associated dual semigroups.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
T. Alvarez and M. González,Some examples of tauberian operators, Proc. Amer. Soc.111 (1991) 1023–1027.
S. Buehler,The James function space, Doctoral Thesis, University of Texas at Austin, 1994.
P.G. Casazza and T.J. Shura,Tsirelson space, Lecture Notes in Math.1363. Springer, 1989.
W.J. Davis, T. Figiel, W.B. Johnson and A. Pelczyński,Factoring weakly compact operators, J. Funct. Anal.17 (1974), 311–327.
J. Diestel,Sequences and series in Banach spaces. Springer, 1984.
D.P. Giesy and R.C. James,Uniformly non-ℓ (1) and B-convex Banach spaces, Studia Math.48 (1973), 61–69.
M. González and A. Martínez-Abejón,Supertauberian operators and perturbations, Arch. Math.64 (1995), 423–433.
M. González and A. Martínez-Abejón,Tauberian operators on L 1 (μ) spaces, Studia Math.125 (1997), 289–303.
M. González and A. Martínez-Abejón,Ultrapowers and semi-Fredholm operators, Bollettino U.M.I.11-B (1997), 415–433.
M. González and A. Martínez-Abejón,Lifting unconditionally converging series and semigroups, Bull. Austral. Math. Soc.57 (1998), 135–146.
M. González and A. Martínez-Abejón,Local reflexivity of dual Banach spaces, Pacific J. Math.189 (1999), 263–278.
M. González and V.M. Onieva,Characterizations of tauberian operators and other semigroups of operators, Proc. Amer. Math. Soc.108 (1990), 399–405.
S. Heinrich,Ultraproducts in Banach space theory, J. reine angew. Math.313 (1980), 72–104.
S. Heinrich,Finite representability and super-ideals of operators, Dissertationes Mathematicae162 (1980), 5–37.
R.C. James,Uniformly non-square Banach spaces, Ann. of Math.80 (1964), 542–550.
N. Kalton and A. Wilansky,Tauberian operators on Banach spaces, Proc. Amer. Soc.57 (1976), 251–255.
A. Lebow and M. Schechter,Semigroups of operators and measures of noncompactness, J. Funct. Anal.7, (1971), 1–26.
J. Lindenstrauss and C. Stegall,Examples of separable spaces which do not contain ℓ 1 and whose duals are non-separable, Studia Math.54, (1975), 81–105.
R. Neidinger and H. Rosenthal,Norm-attainment of linear functionals on subspaces and characterizations of tauberian operators, Pacific J. Math.118 (1985), 215–228.
H. Rosenthal,On wide-(s) sequences and their applications to certain classes of operators, Pacific J. Math.189 (1999), 311–338.
W. Schachermayer,For a Banach space isomorphic to its square the Radon-Nikodym property and the Krein-Milman property are equivalent, Studia Math.81 (1985), 329–339.
D.G. Tacon,Generalized semi-Fredholm transformations, J. Austral math. Soc.34 (1983), 60–70.
D.G. Tacon,Generalized Fredholm transformations, J. Austral math. Soc.37 (1984), 89–97.
P. Wojtaszczyk, Banach spaces for analysts. Cambridge University Press, 1991.
Author information
Authors and Affiliations
Additional information
Supported in part by DGICYT Grant PB 94-1052 (Spain).
Supported by a postdoctoral Grant of the Ministry of Spain for Education and Science
Rights and permissions
About this article
Cite this article
Gonzalez, M., Martinez-Abejon, A. Ultrapowers and semigroups of operators. Integr equ oper theory 37, 32–47 (2000). https://doi.org/10.1007/BF01673621
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01673621