Publication: Best possible Compactness Results of Lions-Peetre Type
Loading...
Files
Full text at PDC
Publication Date
2001
Authors
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Cambridge Univ Press
Citations
Exportar
Abstract
If T : A0 ! B boundedly and T : A1 ! B compactly, then a result of Lions{Peetre shows that T : A ! B compactly for a certain class of spaces A which are intermediate with respect to A0 and A1. We investigate to what extent such results can hold for arbitrary intermediate spaces A. The `dual' case of an operator S such that S : X ! Y0 boundedly and S : X ! Y1 compactly, is also considered,
as well as similar questions for other closed operator ideals.
Description
UCM subjects
Unesco subjects
Keywords
Citation
C. Bennett and R. Sharpley, Interpolation of operators (Academic Press, New York,
1988).
J. Bergh and J. L¨ofstr¨om, Interpolation spaces. An introduction, Grundlehren der
Mathematische Wissenschaften, vol. 223 (Springer, Berlin, 1976).
O. J. Beucher, On interpolation of strictly (co-)singular linear operators, Proc. R. Soc.
Edinb. A112 (1989), 263{269.
Y. Brudnyi and N. Krugljak, Interpolation functors and interpolation spaces, vol. 1
(North Holland, Amsterdam, 1991).
F. Cobos, T. K¨uhn and T. Schonbek, One-sided compactness results for Aronszajn{
Gagliardo functors, J. Funct. Analysis 106 (1992), 274{313.
F. Cobos, A. Manzano and A. Mart_inez, Interpolation theory and measures related
to operator ideals, Q. J. Math. Oxford (2) 50 (1999), 401{416.
F. Cobos and J. Peetre, Interpolation of compactness using Aronszajn{Gagliardo functors,
Israel J. Math. 68 (1989), 220{240.
J. Diestel, H. Jarchow and A. Tonge, Absolutely summing operators, Cambridge
Studies in Advanced Mathematics, vol. 43 (Cambridge University Press, 1995).
A. A. Dmitriev, The interpolation of one-dimensional operators, Vorone_z Gos. Univ.
Trudy Nau_cn.-Issled. Inst. Mat. VGU Vyp. 11 Sb. Statej Funkcional. Anal. i Prilozen 11
(1973), 31{43 (in Russian).
E. Gagliardo, A uni_ed structure in various families of function spaces. Compactness
and closure theorems, in Proc. Int. Symp. Linear Spaces, Jerusalem 1960, pp. 237{241
(Pergamon, Oxford, 1961).
G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals (I), Math.
Z. 27 (1928), 565{606.
S. Heinrich, Closed operator ideals and interpolation, J. Funct. Analysis 35 (1980),
397{411.
M. A. Krasnoselskii, P. P. Zabreiko, E. I. Pustylnik and P. E. Sobolevskii,
Integral operators in spaces of summable functions (Noordho_, Leyden, 1976).
J. L. Lions and J. Peetre, Sur une classe d'espaces d'interpolation, Inst. Hautes Etudes
Sci. Publ. Math. 19 (1964), 5{68.
M. Masty lo, On interpolation of compact operators, Funct. Approx. Comment. Math.
26 (1998), 293{311.
A. Persson, Compact linear mappings between interpolation spaces, Ark. Mat. 5 (1964),
215{219.
A. Pietsch, Operator ideals (North-Holland, Amsterdam, 1980).
E. I. Pustylnik, On optimal interpolation and some interpolation properties of Orlicz
spaces, Soviet. Math. Dokl. 27 (1983), 333{336. (Translation from Dokl. Akad. Nauk SSSR
269 (1983), 292{295.)
E. I. Pustylnik, Embedding functions and their role in interpolation theory, Abstract
Appl. Analysis 1 (1996), 305{325.
M. F. Teixeira and D. E. Edmunds, Interpolation theory and measures of non-compactness,
Math. Nachr. 104 (1981), 129{135.