Lattice structures on some Banach spaces

Mastyło, Mieczysław

doi:10.1090/S0002-9939-2011-11151-1

Lattice structures on some Banach spaces
HTML articles powered by AMS MathViewer

by Mieczysław Mastyło PDF

Proc. Amer. Math. Soc. 140 (2012), 1413-1422 Request permission

Abstract:

The purpose of this paper is to study Banach lattice constants $d_n$ and $e_n$ originally introduced by Kalton. We prove an interpolation theorem for positive operators and derive estimates of the lattice constants for Banach lattices generated by positive interpolation functors. In particular, we provide estimates of these constants for Calderón-Loznanovskii spaces. We also find the lattice constants for Marcinkiewicz and Lorentz spaces. As applications, we prove results concerning lattice structures of studied spaces.

References

Charalambos D. Aliprantis and Owen Burkinshaw, Positive operators, Pure and Applied Mathematics, vol. 119, Academic Press, Inc., Orlando, FL, 1985. MR 809372
Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802
Yu. A. Brudnyĭ and N. Ya. Krugljak, Interpolation functors and interpolation spaces. Vol. I, North-Holland Mathematical Library, vol. 47, North-Holland Publishing Co., Amsterdam, 1991. Translated from the Russian by Natalie Wadhwa; With a preface by Jaak Peetre. MR 1107298
A. V. Bukhvalov, Theorems on interpolation of sublinear operators in spaces with mixed norm, Qualitative and approximate methods for investigating operator equations, Yaroslav. Gos. Univ., Yaroslavl′1984, pp. 90–105, 130 (Russian). MR 841773
A.-P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190. MR 167830, DOI 10.4064/sm-24-2-113-190
W. B. Johnson, B. Maurey, G. Schechtman, and L. Tzafriri, Symmetric structures in Banach spaces, Mem. Amer. Math. Soc. 19 (1979), no. 217, v+298. MR 527010, DOI 10.1090/memo/0217
N. J. Kalton, Lattice structures on Banach spaces, Mem. Amer. Math. Soc. 103 (1993), no. 493, vi+92. MR 1145663, DOI 10.1090/memo/0493
S. G. Kreĭn, Yu. Ī. Petunīn, and E. M. Semënov, Interpolation of linear operators, Translations of Mathematical Monographs, vol. 54, American Mathematical Society, Providence, R.I., 1982. Translated from the Russian by J. Szűcs. MR 649411
Thomas Kühn and Mieczysław Mastyło, Weyl numbers and eigenvalues of abstract summing operators, J. Math. Anal. Appl. 369 (2010), no. 1, 408–422. MR 2643879, DOI 10.1016/j.jmaa.2010.03.043
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367
G. Ja. Lozanovskiĭ, Certain Banach lattices. III, IV, Sibirsk. Mat. Ž. 13 (1972), 1304–1313, 1420; ibid. 14 (1973), 140–155, 237 (Russian). MR 0336314
G. Ya. Lozanovskiĭ, Transformations of ideal Banach spaces by means of concave functions, Qualitative and approximate methods for the investigation of operator equations, No. 3 (Russian), Yaroslav. Gos. Univ., Yaroslavl′1978, pp. 122–148 (Russian). MR 559326
V. I. Ovchinnikov, The method of orbits in interpolation theory, Math. Rep. 1 (1984), no. 2, i–x and 349–515. MR 877877
Shlomo Reisner, On two theorems of Lozanovskiĭ concerning intermediate Banach lattices, Geometric aspects of functional analysis (1986/87), Lecture Notes in Math., vol. 1317, Springer, Berlin, 1988, pp. 67–83. MR 950976, DOI 10.1007/BFb0081736
Vladimir A. Shestakov, Transformations of Banach ideal spaces and interpolation of linear operators, Bull. Acad. Polon. Sci. Sér. Sci. Math. 29 (1981), no. 11-12, 569–577 (1982) (Russian, with English summary). MR 654216