### Monomial decomposition of homogeneous polynomials in vector lattices

#### Abstract

‎ ‎The paper is devoted to the characterization and weighted shift representation of regular homogeneous polynomials between vector lattices admitting a decomposition into a sum of monomials in lattice homomorphisms‎. ‎The main tool is the factorization theorem for order bounded disjointness preserving multilinear operators obtained earlier by the authors‎.

#### Article information

Source
Adv. Oper. Theory, Volume 4, Number 2 (2019), 428-446.

Dates
Accepted: 26 September 2018
First available in Project Euclid: 1 December 2018

https://projecteuclid.org/euclid.aot/1543633236

Digital Object Identifier
doi:10.15352/aot.1807-1394

Mathematical Reviews number (MathSciNet)
MR3883145

Zentralblatt MATH identifier
07009318

Subjects
Secondary: 46G25‎ ‎47A40

#### Citation

‎Kusraev, Anatoly G‎.; ‎Kusraeva, Zalina A‎. Monomial decomposition of homogeneous polynomials in vector lattices. Adv. Oper. Theory 4 (2019), no. 2, 428--446. doi:10.15352/aot.1807-1394. https://projecteuclid.org/euclid.aot/1543633236

#### References

• C. D. Aliprantis and O. Burkinshaw, Positive operators, Acad. Press Inc., London etc., 1985.
• F. Ben Amor, Orthogonally additive homogenous polynomials on vector lattices, Commun. Alg. 43 (2015), no. 3, 1118–1134.
• C. B. Bernau, C. B. Huijsmans, B. de Pagter, Sums of lattice homomorphisms, Proc. Amer. Math. Soc. 115 (1992), no. 1, 151–156.
• K. Boulabiar, Products in almost $f$-algebras, Comment. Math. Univ. Carolin. 41 (2000), no. 4, 747–759.
• K. Boulabiar, Some aspects of Riesz multimorphisms, Indag. Mathem., N.S. 13(2002), no. 4, 419–432.
• K. Boulabiar and G. Buskes, Vector lattice powers: $f$-algebras and functional calculus, Comm. Algebra. 34 (2006), no. 4, 1435–1442.
• Q. Bu, G. Buskes, Polynomials on Banach lattices and positive tensor products, J. Math. Anal. Appl. 388 (2012), 845–862.
• G. Buskes and A. G. Kusraev, Extension and representation of orthoregular maps, Vladikavkaz Mat. Zh. 9 (2007), no. 1, 16–29.
• G. Buskes and A. van Rooij, Almost $f$-algebras: commutativity and the Cauchy–Schwarz inequality, Positivity. 4 (2000), no. 1, 227–231.
• G. Buskes and A. van Rooij, Squares of Riesz spaces, Rocky Mountain J. Math. 31 (2001), no. 1, 45-56.
• G. Buskes and A. van Rooij, Bounded variation and teensor products of Banach lattices, Positivity 7 (2003), no. 1/2, 47-59.
• D. C. Carothers and W. A. Feldman,Sums of homomorphisms on Banach lattices, J. Operator theory 24 (1990), 337–349.
• S. Dineen, Complex analysis on infinite dimensional spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 1999.
• A. E. Gutman, Banach bundles in the theory of lattice-normed spaces, Linear operators compatible with order [in Russian],63–211, Sobolev Institute Press, Novosibirsk, 1995.
• A. E. Gutman, Disjointness preserving operators, Vector Lattices and Integral Operators /Ed. S. S. Kutateladze. 361–454, Dordrecht etc.: Kluwer, 1996.
• A. Ibort, P. Linares, and J. G. Llavona, A representation theorem for orthogonally additive polynomials on Riesz spaces, arXiv: 1203.2379v1.
• A. G. Kusraev and S. S. Kutateladze, Subdifferentials: Theory and application, Kluwer Academic Publ., Dordrecht, 1995.
• A. G. Kusraev and S. S. Kutateladze, Boolean valued analysis: Selected topics, Vladikavkaz, Vladikavkaz Scientific Center Press, 2014.
• A. G. Kusraev and S. N. Tabuev, On disjointness preserving bilinear operators, Vladikavkaz Mat. Zh. 6 (2004), no. 1, 58–70.
• A. G. Kusraev and S. N.Tabuev, On multiplicative representation of disjointness preserving bilinear operators, Sib. Math. J. 49 (2008), no. 2, 357–366.
• A. G. Kusraev and Z. A. Kusraeva, Factorization of order bounded disjointness preserving multilinear operators, Springer Proceedings in Mathematics & Statistics (to appear).
• A. G. Kusraev, Z. A. Kusraeva, Sums of order bounded disjointness preserving multilinear operators, Sib. Math. J. (to appear).
• Z. A. Kusraeva, Representation of orthogonally additive polynomials, Sib. Math. J. 52 (2011), no. 2, 248–255.
• Z. A. Kusraeva, Orthogonal additive polynomials on vector lattices, Thesis, Sobolev Inst. of Math. of the Sib. Branch of the RAS, Novosibirsk, 2013.
• Z. A. Kusraeva, Characterization and multiplicative representation of homogeneous disjointness preserving polynomials, Vladikavkaz Mat. Zh. 18 (2016), no. 1, 51–62.
• P. Linares,Orthogonal additive polynomials and applications, Thesis, Departamento de Analisis Matematico Universidad Complutense de Madrid, 2009.
• J. Loane, Polynomials on Riesz spaces, Thesis. Department of Math. Nat. Univ. of Ireland, Galway, 2007.
• V. A. Radnaev, On $n$-disjoint operators, Siberian Adv. Math. 7 (1997), no. 4, 44–78.
• V. A. Radnaev, On metric $n$-indecomposability in ordered lattice normed spaces and its applications, PhD Thesis, Sobolev Institute Press, Novosibirsk, 1997.
• J. Szulga, $(p,r)$-convex functions on vector lattices, Proc. Edinburgh Math. Soc. (2) 37 (1994), no. 2, 207–226.
• V. G. Troitsky and O. Zabeti, Fremlin tensor products of concavifications of Banach lattices, Positivity 18 (2014), no. 1, 191–200.
• M. A. Toumi, Orthogonally additive polynomials on Dedekind $\sigma$-complete vector lattices, Math. Proc. R. Ir. Acad. 110A (2010), no. 1, 83–94.