Abstract
The main results obtained are:
– A Dedekind complete Banach lattice Y has a Fatou norm if and only if, for any Banach lattice X, the regular-norm unit ball Ur = {T ∈ Lr(X,Y): ||T||r ≤ 1} is closed in the strong operator topology on the space of all regular operators, Lr(X,Y).
– A Dedekind complete Banach lattice Y has a norm which is both Fatou and Levi if and only if, for any Banach lattice X, the regular-norm unit ball Ur is closed in the strong operator topology on the space of all bounded operators, L(X,Y).
– A Banach lattice Y has a Fatou–Levi norm if and only if for every L-space X the space L(X,Y) is a Banach lattice under the operator norm.
– A Banach lattice Y is isometrically order isomorphic to C(S) with the supremum norm, for some Stonean space S, if and only if, for every Banach lattice X, L(X,Y) is a Banach lattice under the operator norm.
Several examples demonstrating that the hypotheses may not be removed, as well as some applications of the results obtained to the spaces of operators are also given. For instance:
– If X = Lp(μ) and Y = Lq(ν), where 1 < p,q < ∞, then Lr(X,Y) is a first category subset of L(X,Y).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramovich, Y. A.: 1972, On some properties of norm in normed lattices and their maximal normed extensions. Thesis, Leningrad State Univ.
Abramovich, Y. A.: 1977, ‘Some theorems on normed lattices’. Vestnik Leningrad Univ. 4, 153–159.
Abramovich, Y. A.: 1978, ‘Some new characterizations of AM-spaces’. Ann. Univ. Craiova 6, 15–26.
Abramovich, Y. A.: 1990, ‘When each continuous operator is regular’. Funct. Analysis, Optimization and Math. Economics, Oxford Univ. Press, 133–140.
Abramovich, Y. A. and L. P. Janovsky: 1982, ‘Applications of the Rademacher systems to operator characterizations of Banach lattices’. Colloq. Math. 46, 75–78.
Abramovich, Y. A. and A. W. Wickstead: 1992, ‘A compact regular operator without modulus’. Proc. Amer. Math. Soc. 116, 721–726.
Abramovich, Y. A. and A. W. Wickstead: 1995, ‘Solutions of several problems in the theory of compact positive operators’. Proc. Amer. Math. Soc. 123, 3021–3026.
Abramovich, Y. A. and A. W. Wickstead: (forthcoming), ‘When each continuous operator is regular, II’. Indag. Math.
Aliprantis, C. D. and O. Burkinshaw: 1985, Positive Operators, Academic Press, New York & London.
Amemiya I.: 1953, ‘A generalization of Riesz-Fischer's theorem’. J. Math. Soc. Japan 5, 353–354.
Arendt, W. and J. Voigt: 1991, ‘Approximation of multipliers by regular operators’. Indag. Math. 2, 159–170.
Cartwright, D. I. and H. P. Lotz: 1975, ‘Some characterizations of AM-and AL-spaces’. Math. Z. 142, 97–103.
Fremlin, D. H.: 1975, ‘A positive compact operator’. Manuscripta Math. 15, 323–327.
Huijsmans, C. B. and A. W. Wickstead: 1996, ‘Bands in Lattices of Operators’. Proc. Amer. Math. Soc. 124, 3835–3841.
Kantorovich, L. V. and B. Z. Vulikh: 1937, ‘Sur la représentation des opérations linéares’. Compos. Math. 5, 119–165.
Meyer-Nieberg, P.: 1991, Banach Lattices, Springer-Verlag, Berlin, Heidelberg, New York.
Synnatzschke J.: 1972, ‘On the conjugate of a regular operator and some applications to the questions of complete continuity of regular operators’. Vestnik Leningr. Univ. Mat. Meh. Astronom. n. 1, 60–69.
Vulikh, B. Z.: 1967, ‘Introduction to the theory of partially ordered spaces’. Wolters-Noordhoff Sci. Publication, Groningen.
Voigt J.: 1988, ‘The projection onto the center of operators in a Banach lattice’. Math. Zeitschr. 199, 115–117.
Wickstead A.W.: 1990, ‘The regularity of order bounded operators into C(K)’. Quart. J. Math. Oxford (2) 41, 359–368.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Abramovich, Y.A., Chen, Z.L. & Wickstead, A.W. Regular-Norm Balls can be Closed in the Strong Operator Topology. Positivity 1, 75–96 (1997). https://doi.org/10.1023/A:1009743831883
Issue Date:
DOI: https://doi.org/10.1023/A:1009743831883