Abstract
In this note we construct a pair of Banach lattices X and Y, which have the following properties:
-
a)
X is not order isomorphic to an AL-space,
-
b)
Y is not order isomorphic to an AM-space,
-
c)
for any continuous linear operator T:X → Y there exists a modulus ¦T¦: X → Y.
This example refutes the conjecture of Cartwright-Lotz, saying that the negation of at least one of the conditions a) or b) is necessary for the validity of c).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature cited
L. V. Kantorovich and B. Z. Vulikh, Compos. Math.,5, 119–165 (1937).
B. Z. Vulikh, Introduction to the Theory of Partially Ordered Spaces, Gordan and Breach (1967).
D. Fremlin, Indag. Math.,36, 270–275 (1974).
D. I. Cartwright and H. P. Lotz, Math. Z.,142, 97–103 (1975).
P. Orno, Israel J. Math.,19, No. 3, 264–265 (1974).
Yu. A. Abramovich, Vestn. Leningr. Gos. Univ., No. 13, 5–11 (1971).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 73, pp. 188–192, 1977.
Rights and permissions
About this article
Cite this article
Abramovich, Y.A. Space of operators acting from one banach lattice to another. J Math Sci 34, 2134–2137 (1986). https://doi.org/10.1007/BF01741586
Issue Date:
DOI: https://doi.org/10.1007/BF01741586