Hostname: page-component-f554764f5-nqxm9 Total loading time: 0 Render date: 2025-04-11T18:10:43.606Z Has data issue: false hasContentIssue false
  • English
  • Français

Measure of weak noncompactness and real interpolation of operators

Published online by Cambridge University Press:  17 April 2009

Andrzej Kryczka ,
Stanislaw Prus  and
Mariusz Szczepanik
Andrzej Kryczka
Affiliation:
Institute of Mathematics, Maria Curie-Skłodowska University, 20–031 Lublin, Poland e-mail: [email protected]@[email protected]
Stanislaw Prus
Affiliation:
Institute of Mathematics, Maria Curie-Skłodowska University, 20–031 Lublin, Poland e-mail: [email protected]@[email protected]
Mariusz Szczepanik
Affiliation:
Institute of Mathematics, Maria Curie-Skłodowska University, 20–031 Lublin, Poland e-mail: [email protected]@[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A new measure of weak noncompactness is introduced. A logarithmic convexity-type result on the behaviour of this measure applied to bounded linear operators under real interpolation is proved. In particular, it gives a new proof of the theorem showing that if at least one of the operators T: Ai → Bi, i = 0, 1 is weakly compact, then so is T : Aθ,p → Bθ,p for all 0 < θ < 1 and 1 < P < ∞.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 62 , Issue 3 , December 2000 , pp. 389 - 401
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Akhmerov, R.R., Kamenskii, M.I., Potapov, A.S., Rodkina, A.E. and Sadovskii, B.N., Measures of noncompactness and condensing operators (Birkhäuser Verlag, Basel, Boston, Berlin 1992).Google Scholar
[2]Aksoy, A.G. and Khamsi, M.A., Nonstandard methods in fixed point theory (Springer-Verlag, Berlin, Heidelberg, New York, 1990).Google Scholar
[3]Aksoy, A.G. and Maligranda, L., ‘Real interpolation and measure of weak noncompactness’, Math. Nachr. 175 (1995), 5–12.CrossRefGoogle Scholar
[4]Astala, K. and Tylli, H.-O., ‘Seminorms related to weak compactness and to Tauberian operators’ Math. Proc. Cambribge Philos. Soc. 107 (1990), 367–375.CrossRefGoogle Scholar
[5]Banaś, J., ‘Applications of Measures of weak noncompactness and some classes of operators in the theory of functional equations in the Lebesgue space’,(Proceedings of the Second World Congress of Nonlinear Analysts, Part 6,Athens,1996),Nonlinear Anal. 30 (1997), 3283–3293.CrossRefGoogle Scholar
[6]Banaś, J. and Martinón, A., ‘Measures of weak noncompactness in Banach sequence spaces’, Portugal. Math. 52 (1995), 131–138.Google Scholar
[7]Banaś, J. and Rivero, J., ‘On measures of weak noncompactness’, Ann. Mat. Pura Appl. 151, 213–224.Google Scholar
[8]Beauzamy, B., Espaces d'interpolation réels: tapologie et géométrie (Springer-Verlag, Berlin, Heidelberg, New York, 1978).CrossRefGoogle Scholar
[9]Brudnyi, Yu.A. and Krugljak, N. Ya., Interpolation functors and interpolation spaces. Vol. I (North-Holland, Amsterdam, New York, Oxford, Tokyo, 1991).Google Scholar
[10]Cobos, F., Interpolation theory and measures related to operator ideals, (Krbec, M. and Kufner, A., Editors), Nonlinear analysis, function spaces and applications, Vol. 6 (Olympia Press, Prague, 1999).Google Scholar
[11]Cobos, F., Fernández-Martínez, P. and Martínez, A., ‘Interpolation of the measure of non-compactness by the real method’, Studia Math. 135 (1999), 25–38.Google Scholar
[12]Cobos, F., Manzano, A. and Martínez, A., ‘Interpolation theory and measures related to operator ideals’, Quart. J. Math. Oxford Ser. (2) 50 (1999), 401–416.CrossRefGoogle Scholar
[13]Cobos, F. and Martínez, A., ‘Extreme estimates for interpolated operators by the real method’, J. London Math. Soc. (2) 60 (1999), 860–870.Google Scholar
[14]Cobos, F. and Martínez, A., ‘Remarks on interpolation properties of the measure of weak non-compactness and ideal variations’, Math. Nachr. 208 (1999), 93–100.Google Scholar
[15]Cwikel, M., ‘Real and complex interpolation and extrapolation of compact operators’, Duke Math. J. 65 (1992), 333–343.Google Scholar
[16]De Blasi, F.S., ‘On a property of the unit sphere in a Banach space’, Bull. Math. Soc. Sci. Math. R.S. Roumanie 21 (1977), 259–262.Google Scholar
[17]Van Dulst, D., Reflexive and superreflexive Banach spaces (Mathematisc Centrum, Amsterdam, 1978).Google Scholar
[18]Heinrich, S., ‘Closed operator ideals and interpolation’, J. Funct. Anal. 35 (1980), 397–411.Google Scholar
[19]James, R.C., ‘Weak compactness and reflexity’, Israel J. Math. 2 (1964), 101–119.CrossRefGoogle Scholar
[20]Krasnoselskii, M.A. ‘On a theorem of M. Riesz’, (Russian), Dokl. Akad. Nauk SSSR 131 (1960), 246–248.Google Scholar
[21]Lions, J.-L. and Peetre, J., ‘Sur une classe d'espaces d'interpolation’, Inst. Hautes Études Sci. Publ. Math. 19 (1964), 5–68.Google Scholar
[22]Maligranda, L. and Quevedo, A., ‘Interpolation of weakly compact operators’, Arch. Math. 55 (1990), 280–284.Google Scholar
[23]Mastylo, M., ‘On interpolation of weakly compact operators’, Hokkaido Math. J. 22 (1993), 105–114.CrossRefGoogle Scholar
[24]Mazur, S., ‘Über konvexe Mengen in linearen normierten Räumen’, Studia Math. 4 (1933), 70–84.CrossRefGoogle Scholar
[25]Milman, D.P. and Milman, V.D., ‘The geometry of imbeddings with empty intersection. The structure of the unit sphere in a non-reflexive space’, (Russian), Mat. Sbornik 66 (1965), 109–118.Google Scholar
[26]Milman, V.D., ‘Geometric theory of Banach spaces. II. Geometry of the unit ball’, (Russian), Uspekhi Mat. Nauk 26 (1971), 73–149. English translation: Russian Math. Surv. 26 (1971), 79–163.Google Scholar
[27]Teixeira, M.F. and Edmunds, D.E., ‘Interpolation theory and measures of non-compactness’, Math. Nachr. 104 (1981), 129–135.CrossRefGoogle Scholar