Abstract
During the last few years the authors have studied extensively the invariant subspace problem of positive operators; see [6] for a survey of this investigation. In [4] the authors introduced the class of compact-friendly operators and proved for them a general theorem on the existence of invariant subspaces. It was then asked if every positive operator is compact-friendly. In this note, we present an example of a positive operator which is not compact-friendly but which, nevertheless, has a non-trivial closed invariant subspace.
In the process of presenting this example, we also characterize the multiplication operators that commute with non-zero finite-rank operators. We show, among other things, that a multiplication operator M ϕ commutes with a non-zero finite-rank operator if and only the multiplier function ϕ is constant on some non-empty open set.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Y. A. Abramovich, Multiplicative representation of operators preserving disjointness, Netherl. Acad. Wetensch. Proc. Ser. A 86(1983), 265-279.
Y. A. Abramovich, C. D. Aliprantis and O. Burkinshaw, Positive operators on Krein spaces, Acta Appl. Math. 27(1992), 1-22.
Y. A. Abramovich, C. D. Aliprantis and O. Burkinshaw, Invariant subspaces of operators on ℓp-spaces, J. Funct. Anal. 115(1993), 418-424.
Y. A. Abramovich, C.D. Aliprantis and O. Burkinshaw, Invariant subspaces of positive operators, J. Funct. Anal. 124(1994), 95-111.
Y. A. Abramovich, C. D. Aliprantis and O. Burkinshaw, Local quasinilpotence, cycles and invariant subspaces, in: N. Kalton, E. Saab, and S. Montgomery-Smith eds., Interaction Between Functional Analysis, Harmonic Analysis, and Probability, Lecture Notes in Pure and Applied Mathematics, vol. 175, Dekker, New York, 1996, pp. 1-12.
Y. A. Abramovich, C. D. Aliprantis and O. Burkinshaw, The invariant subspace problem: some recent advances, Rend. Instit. Mat. Univ. Trieste, forthcoming.
C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis, Studies in Economic Theory, # 4, Springer Verlag, Heidelberg-New York, 1994.
C. D. Aliprantis and O. Burkinshaw, Positive Operators, Academic Press, New York-London, 1985.
D. W. Hadwin, An operator still not satisfying Lomonosov's hypothesis, Proc. Amer. Math. Soc. 123(1995), 3039-3041.
D. W. Hadwin, E. A. Nordgren, H. Radjavi and P. Rosenthal, An operator not satisfying Lomonosov's hypothesis, J. Funct. Anal. 38(1980), 410-415.
V. I. Lomonosov, Invariant subspaces of the family of operators that commute with a completely continuous operator, Funktsional. Anal. i Prilozhen 7(1973), No. 3, 55-56. (Russian)
P. Meyer-Nieberg, Banach Lattices, Springer Verlag, Heidelberg-New York, 1991.
B. de Pagter, Irreducible compact operators, Math. Z. 192(1986), 149-153.
Rights and permissions
About this article
Cite this article
Abramovich, Y.A., Aliprantis, C.D. & Burkinshaw, O. Multiplication and Compact-friendly Operators. Positivity 1, 171–180 (1997). https://doi.org/10.1023/A:1009781922898
Issue Date:
DOI: https://doi.org/10.1023/A:1009781922898