Homogeneous polynomials, root mean power, and geometric means in vector lattices

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.23671/VNC.2014.4.10260 Homogeneous polynomials, root mean power, and geometric means in vector lattices Kusraeva, Z. A. Vladikavkaz Mathematical Journal 2014. Vol. 16. Issue 4. Abstract: It is proved that for a homogeneous orthogonally additive polynomial \(P\) of degree \(s\in\mathbb{N}\) from a uniformly complete vector lattice \(E\) to some convex bornological space the equations \(P(\mathfrak{S}_s(x_{1},\ldots,x_{N}))= P(x_{1})+\ldots+P(x_{N})\) and \(P(\mathfrak{G}(x_{1},\ldots,x_{s}))= \check{P}(x_{1},\ldots,x_{s})\) hold for all positive \(x_{1},\ldots,x_{s}\in E\), where \(\check{P}\) is an \(s\)-linear operator generating \(P\), while \(\mathfrak{S}_s(x_{1},\ldots,x_{N})\) and \(\mathfrak{G}(x_{1},\ldots,x_{s})\) stand respectively for root mean power and geometric mean in the sense of homogeneous functional calculus. Keywords: vector lattice, homogeneous polynomial, linearization of a polynomial, root mean power, geometric mean Language: Russian Download the full text For citation: Kusraeva Z. A. Homogeneous polynomials, root mean power, and geometric means in vector lattices // Vladikavkazskii matematicheskii zhurnal [Vladikavkaz Math. J.], vol. 16, no. 4, pp. 49-53. DOI 10.23671/VNC.2014.4.10260 + References 1. Aliprantis C. D., Burkinshaw O. Positive Operators, N. Y., Acad. Press, 1985. xvi+367 p. 2. Benyamini Y., Lassalle S., Llavona J. G. Homogeneous orthogonally additive polynomials on Banach lattices. Bull. London Math. Soc., 2006, vol. 38, no. 3. pp. 459-469. 3. Boulabiar K., Buskes G. Vector lattice powers: f-algebras and functional Calculus. Comm. Algebra, 2006, vol. 34, no. 4, pp. 1435-1442. 4. Bu Q., Buskes G., Kusraev A. G. Bilinear maps on product of vector lattices: A survey. Positivity, Eds. K. Boulabiar, G. Buskes, A. Triki. Basel a.o., Birkhauser, 2007, pp. 97-126. 5. Buskes, G., de Pagter.B., van Rooij. A. Functional calculus on Riesz spaces. Indag. Math. (N.S.), 1991, vol. 4, no. 2, pp. 423-436. 6. Meyer-Nieberg P. Banach Lattices. Berlin etc., Springer, 1991, 395 p. 7. Kusraeva Z. A. Representation of orthogonally additive polynomials. Siberian Math. J., 2011, vol. 52, no. 2, pp. 248-255. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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