×

Factorization by lattice homomorphisms. (English) Zbl 0558.47029

The main theorem is the following theorem which generalizes a Radon- Nikodym type theorem due to W. A. J. Luxemburg and A. R. Schep [Indag. Math. 40, 357-375 (1978; Zbl 0389.47018)].
Let E, F, G be Banach lattices, E order complete and \(V: F\to G\) be a lattice homomorphism. Given a positive linear mapping \(S: G\to E\), every positive linear \(T: F\to E\) satisfying \(T\leq S\circ V\) admits a factorization \(T=S_ 1\circ V\) where \(S_ 1: G\to E\) is linear and \(0\leq S_ 1\leq S\). Applications to injective Banach lattices and spectral theory are given.

MSC:

47B60 Linear operators on ordered spaces
46B42 Banach lattices
47A10 Spectrum, resolvent
46M10 Projective and injective objects in functional analysis

Citations:

Zbl 0389.47018
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Aliprantis, C.D., Burkinshaw, O., Kranz, P.: On lattice properties of the composition operator. Manuscripta Math.36, 19-31 (1981) · Zbl 0478.46008 · doi:10.1007/BF01174810
[2] Arendt, W.: On the spectrum of regular operators. Dissertation, Tübingen 1979
[3] Arendt, W.: On the o-spectrum of regular operators and the spectrum of measures. Math. Z.178, 271-287 (1981) · Zbl 0468.47022 · doi:10.1007/BF01262044
[4] Bonsall, F.F., Duncan, J.: Complete Normed Algebras. Berlin-Heidelberg-New York: Springer 1973 · Zbl 0271.46039
[5] Cartwright, D.I.: Extension of positive operators between Banach lattices. Mem. Amer. Math. Soc.164 (1975) · Zbl 0314.47015
[6] Haid, W.: Sätze vom Radon-Nikodym-Typ für Operatoren auf Banachverbänden. Semesterbericht Funktionalanalysis, Tübingen, Wintersemester 1981/82 · Zbl 0526.47009
[7] Hart, D.R.: Disjointness preserving oprators. Thesis, Pasadena 1983
[8] Haydon, R.: Injective Banach lattices. Math. Z.156, 19-47 (1977) · Zbl 0353.46003 · doi:10.1007/BF01215126
[9] Lindenstrauss, J., Tzafriri, L.: On the isomorphic classification of injective Banach lattices. Math. Analysis and Applications, Part B, Advances in Math. Suppl. Studies, Vol. 7 B, pp. 489-498. New York: Academic Press 1981 · Zbl 0478.46019
[10] Lotz, H.P.: Extensions and liftings of positive linear mappings on Banach lattices. Trans. Amer. Math. Soc.211, 85-100 (1975) · Zbl 0351.47005 · doi:10.1090/S0002-9947-1975-0383141-7
[11] Luxemburg, W.A.J.: Some Aspects of the Theory of Riesz Spaces. Univ. Arkansas Lecture Notes in Math.4. Fayetteville: Univ. of Arkansas 1979 · Zbl 0431.46003
[12] Luxemburg, W.A.J., Schep, A.R.: A Radon-Nikodym type theorem for positive operators and a dual. Indian Math.40, 357-375 (1978) · Zbl 0389.47018
[13] Schaefer, H.H.: Banach Lattices and Positive Operators. Berlin-Heidelberg-New York: Springer 1974 · Zbl 0296.47023
[14] Schaefer, H.H.: On the o-spectrum of order bounded operators. Math. Z.154, 79-84 (1977) · Zbl 0344.47019 · doi:10.1007/BF01215115
[15] Schaefer, H.H.: Aspects of Banach lattices. In: MAA Stud. Math. (R.C. Bartle, ed.). Washington: Math. Asoc. of America 1980 · Zbl 0494.46021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.