Linear polarization constants of Hilbert spaces

Pappas, A; Revesz, SGy

dc.contributor.author	Pappas, A	en
dc.contributor.author	Revesz, SGy	en
dc.date.accessioned	2014-03-01T01:20:43Z
dc.date.available	2014-03-01T01:20:43Z
dc.date.issued	2004	en
dc.identifier.issn	0022-247X	en
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/16023
dc.subject	Approximation Order	en
dc.subject	banach space	en
dc.subject	Hilbert Space	en
dc.subject	Linear Algebra	en
dc.subject	Lower Bound	en
dc.subject.classification	Mathematics, Applied	en
dc.subject.classification	Mathematics	en
dc.subject.other	PLANK PROBLEM	en
dc.subject.other	LOWER BOUNDS	en
dc.subject.other	POLYNOMIALS	en
dc.subject.other	NORMS	en
dc.title	Linear polarization constants of Hilbert spaces	en
heal.type	journalArticle	en
heal.identifier.primary	10.1016/j.jmaa.2004.06.031	en
heal.identifier.secondary	http://dx.doi.org/10.1016/j.jmaa.2004.06.031	en
heal.language	English	en
heal.publicationDate	2004	en
heal.abstract	This paper has been motivated by previous work on estimating lower bounds for the norms of homogeneous polynomials which are products of linear forms. The purpose of this work is to investigate the so-called nth (linear) polarization constant c(n)(X) of a finite-dimensional Banach space X, and in particular of a Hilbert space. Note that c(n)(X) is an isometric invariant of the space. It has been proved by J. Arias-de-Reyna [Linear Algebra Appl. 285 (1998) 395-408] that if H is a complex Hilbert space of dimension at least n, then c(n)(H) = n(n/2). The same value of c(n)(H) for real Hilbert spaces is only conjectured, but estimates were obtained in many cases. In particular, it is known that the nth (linear) polarization constant of a d-dimensional real or complex Hilbert space H is of the approximate order d(n/2), for n large enough, and also an integral form of the asymptotic quantity c(H), that is the (linear) polarization constant of the Hilbert space H, where dim H = d, was obtained together with an explicit form for real spaces. Here we present another proof, we find the explicit form even for complex spaces, and we elaborate further on the values of c(n)(H) and c (H). In particular, we answer a question raised by J.C. Garcia-Vazquez and R. Villa [Mathematika 46 (1999) 315-322]. Also, we prove the conjectured c(n)(H) = n(n/2) for real Hilbert spaces of dimension n less than or equal to 5. A few further estimates have been also derived. (C) 2004 Elsevier Inc. All rights reserved.	en
heal.publisher	ACADEMIC PRESS INC ELSEVIER SCIENCE	en
heal.journalName	Journal of Mathematical Analysis and Applications	en
dc.identifier.doi	10.1016/j.jmaa.2004.06.031	en
dc.identifier.isi	ISI:000225196300010	en
dc.identifier.volume	300	en
dc.identifier.issue	1	en
dc.identifier.spage	129	en
dc.identifier.epage	146	en