Complutense University Library

Best possible Compactness Results of Lions-Peetre Type

Cobos, Fernando and Cwikel, M. and Matos, P. (2001) Best possible Compactness Results of Lions-Peetre Type. Proceedings of the Edinburgh Mathematical Society, 44 (Part 1). pp. 153-173. ISSN 0013-0915

[img] PDF
Restricted to Repository staff only until 2020.

393kB

Official URL: http://journals.cambridge.org/download.php?file=%2FPEM%2FPEM2_44_01%2FS0013091598001163a.pdf&code=9b3d4

View download statistics for this eprint

==>>> Export to other formats

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.

Item Type:Article
Uncontrolled Keywords:Interpolation of Compact Operators; Rank-One Interpolation Spaces;General Intermediate Spaces; Operator Ideals; Mathematics
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
ID Code:15165
References:

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.

Deposited On:10 May 2012 08:57
Last Modified:06 Feb 2014 10:17

Repository Staff Only: item control page