IJPAM: Volume 113, No. 3 (2017)

Title

CHARACTERIZATION OF k-DISJOINTNESS
PRESERVING NON-LINEAR OPERATORS
BETWEEN BANACH LATTICES

Authors

William Feldman$^1$, Pramod Singh$^2$
$^2$Department of Mathematical Sciences
University of Arkansas
Fayetteville, AR 72701, USA
$^2$Hewlett Packard, Bagmane Tech-Park
Embassy Prime CV Raman Nagar
Bangalore 560093, INDIA

Abstract

A map T, not necessarily linear, between two Banach lattices $E$ and $F$ is said to k-disjointness preserving if $T(f_0) \wedge T(f_1)\wedge \ . \ . \wedge T(f_k) = 0$, whenever $f_0 , f_1 , ..,f_k$ are $k+1$ mutually disjoint positive elements in E and k is the smallest natural number with this property. Certain k-disjointness preserving maps are characterized in terms of cardinality of subsets of $X$, where $C^{\infty}(X)$ is a representation space of $E$. This then facilitates a decomposition of these k-disjoint non-linear operators into a sum of disjointness preserving operators. Further, given $E$ as represented as functions on $X$ and $F$ as functions on $Y$, the disjointness preserving operator $T$ has the property that $Tf(y)= F_y(f(x)$ where $x$ corresponds to $y$ and $F_y$ is an increasing real-valued function defined on the reals or extended reals.

History

Received: November 15, 2016
Revised: January 10, 2016
Published: March 28, 2017

AMS Classification, Key Words

AMS Subject Classification: 46B42,47H07,54G05.
Key Words and Phrases: Disjointness preserving, k-disjointness preserving, extremally disconnected, non-linear operators, Carleman operators, orthogonally additive.

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Bibliography

1
Abramovich, Y.A., Multiplicative representation of disjointness preserving operators, Indag. Math. 45 (1983) 265-279, doi: https://doi.org/10.1016/1385-7258(83)90062-8.

2
Benyamini, Y., Lassalle, S. and Llavona, J., Homogeneous orthogonally additive polynomials on Banach lattices, Bull. London Math. Soc. 38, 3 (2006) 459-469.

3
Bernau, S.J., Huijsmans, C.B. and De Pagter, B., Sums of Lattice Homomorphisms, Proc. Amer Math. Soc. 115, 1 (1992) 151-156, doi: https://doi.org/10.1090/S0002-9939-1992-1086322-8.

4
Carothers, D.C. and Feldman, W.A., Sums of homomorphisms on Banach lattices, J. Operator theory 24 (1990) 337-349

5
Feldman, W.A., Separation properties for Carleman operators on Banach lattices, Positivity 7 (2003) 41-45 doi: https://doi.org/10.1023/A:1025818931354.

6
Feldman, W.A and Singh, P., Positively Decomposable maps between Banach lattices, Positivity 12 (2008) 495-502, doi: https://doi.org/10.1007/s11117-007-2115-5.

7
Grobler, J.J. and Van Eldik, P., Carleman operators on Riesz spaces, Indag. Math. 45 (1983) 421-433, doi: https://doi.org/10.1016/0019-3577-90-90031-H.

8
Pérez-García, David and Ignacio Villanueva, Orthogonally additive polynomials on spaces of continuous functions, J. Math. Anal. Appl. 306, 1 (2005) 97-105, doi: https://doi.org/10.1016/j.jmaa.2004.12.036.

9
Schaefer, H.H., Banach lattices and positive operators, Berlin, Heidelberg, New York: Springer-Verlag (1974), doi: https://doi.org/10.1007/978-3-642-65970-6.

How to Cite?

DOI: 10.12732/ijpam.v113i3.6 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2017
Volume: 113
Issue: 3
Pages: 441 - 453


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