Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-28T05:33:58.271Z Has data issue: false hasContentIssue false
  • English
  • Français

Quantitative approach to weak noncompactness in the polygon interpolation method

Published online by Cambridge University Press:  17 April 2009

Andrzej Kryczka
Andrzej Kryczka
Affiliation:
Institute of Mathematics, Maria Curie-Skłodowska University, 20–031 Lublin, Poland, e-mail: andrzej.kryczka@umcs.lublin.pl
Rights & Permissions [Opens in a new window]

Extract

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.

We study a quantitative approach to weak noncompactness of operators under the Cobos-Peetre polygon interpolation method for Banach N-tuples. In the case of operators acting between two J-spaces or two K-spaces obtained by this method we prove logarithmically convex-type inequalities for certain operator seminorm vanishing on the subspace of weakly compact operators. Geometrically speaking, in these estimates only some triangles inscribed in the polygon are involved. For operators acting from a J-space to a K-space we prove logarithmically convex-type estimates where all polygon vertices are included. In particular, the estimates obtained here give the new proofs of the results showing the relation between distribution of weakly compact operators among polygon vertices and weak compactness of operators under interpolation.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 69 , Issue 1 , February 2004 , pp. 49 - 62
Copyright
Copyright © Australian Mathematical Society 2004

References

[1] Astala, K. and Tylli, H.-O., ‘Seminorms related to weak compactness and to Tauberian operators’, Math. Proc. Camb. Phil. Soc. 107 1990, 367–375.CrossRefGoogle Scholar
[2] Beauzamy, B., Espaces d'interpolation réels: topologie et géométrie (Springer-Verlag, Berlin, Heidelberg, New York, 1978).CrossRefGoogle Scholar
[3] Calderón, A.P., ‘Intermediate spaces and interpolation, the complex method’, Studia Math. 24 1964, 113–190.CrossRefGoogle Scholar
[4] Carro, M.J. and Nikolova, L.I., ‘Interpolation of limited and weakly compact operators on families of Banach spaces’, Acta Appl. Math. 49 1997, 151–177.CrossRefGoogle Scholar
[5] Cobos, F., Cordeiro, J.M. and Martínez, A., ‘Quantitative estimates for interpolated operators by multidimensional methods’, Rev. Math. Complut. 12 1999, 85–103.Google Scholar
[6] Cobos, F., Fernández-Martínez, P. and Martínez, A., ‘On reiteration and the behaviour of weak compactness under certain interpolation methods’, Collect. Math. 50 1999, 53–72.Google Scholar
[7] Cobos, F., Fernández-Martínez, P. and Schonbek, T., ‘Norm estimates for interpolation methods defined by means of polygons’, J. Approx. Theory 80 1995, 321–357.CrossRefGoogle Scholar
[8] Cobos, F., Kühn, T. and Schonbek, T., ‘One-sided compactness results for Aron-szajn-Gagliardo functors’, J. Funct. Anal. 106 1992, 274–313.CrossRefGoogle Scholar
[9] 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
[10] Cobos, F. and Peetre, J., ‘Interpolation of compact operators: the multidimensional case’, Proc. London Math. Soc. 63 1991, 371–400.CrossRefGoogle Scholar
[11] Coifman, R.R., Cwikel, M., Rochberg, R., Sagher, Y. and Weiss, G., ‘A theory of complex interpolation for families of Banach spaces’, Adv. Math. 43 1982, 203–229.CrossRefGoogle Scholar
[12] 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
[13] Fernandez, D.L., ‘Interpolation of 2n Banach spaces’, Studia Math. 65 1979, 175–201.CrossRefGoogle Scholar
[14] Fernandez, D.L., ‘Interpolation of 2d Banach spaces and the Calderón spaces X(E)’, Proc. London Math. Soc. 56 1988, 143–162.CrossRefGoogle Scholar
[15] González, M., Saksman, E. and Tylli, H.-O., ‘Representing non-weakly compact operators’, Studia Math. 113 1995, 265–282.CrossRefGoogle Scholar
[16] Heinrich, S., ‘Closed operator ideals and interpolation’, J. Funct. Anal. 35 1980, 397–411.CrossRefGoogle Scholar
[17] Krein, S.G. and Nikolova, L.I., ‘Holomorphic functions in a family of Banach spaces and interpolation’, (in Russian), Dokl. Akad. Nauk SSSR 250 (1980), 547–550. English translation: Soviet Math. Dokl. 21 1980, 131–134.Google Scholar
[18] Kryczka, A. and Prus, S., ‘Measure of weak noncompactness under complex interpolation’, Studia Math. 147 2001, 89–102.CrossRefGoogle Scholar
[19] Kryczka, A., Prus, S. and Szczepanik, M., ‘Measure of weak noncompactness and real interpolation of operators’, Bull. Austral. Math Soc. 62 2000, 389–401.CrossRefGoogle Scholar
[20] Lions, J.-L. and Peetre, J., ‘Sur une classe d'espaces d'interpolation’, Inst. Hautes Études Sci. Publ. Math. 19 1964, 5–68.CrossRefGoogle Scholar
[21] Maligranda, L., ‘Interpolation between sum and intersection of Banach spaces’, J. Approx. Theory 47 1986, 42–53.CrossRefGoogle Scholar
[22] Maligranda, L., ‘Weakly compact operators and interpolation’, Acta Appl. Math. 27 1992, 79–89.CrossRefGoogle Scholar
[23] Maligranda, L. and Quevedo, A., ‘Interpolation of weakly compact operators’, Arch. Math. 55 1990, 280–284.CrossRefGoogle Scholar
[24] Mastyło, M., ‘On interpolation of weakly compact operators’, Hokkaido Math. J. 22 1993, 105–114.CrossRefGoogle Scholar
[25] Sparr, G., ‘Interpolation of several Banach spaces’, Ann. Mat. Pura Appl. 99 1974, 247–316.CrossRefGoogle Scholar
[26] Tylli, H.-O., ‘Duality of the weak essential norm’, Proc. Amer. Math. Soc. 129 2001, 1437–1443.CrossRefGoogle Scholar
[27] Yoshikawa, A., ‘Sur la théorie d'espaces d'interpolation – les espaces de moyenne de plusieurs espaces de Banach’, J. Fac. Sci. Univ. Tokyo 16 1970, 407–468.Google Scholar