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The real plank problem and some applications.

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Muñoz-Fernández, Gustavo A. and Sarantopoulos, Y and Seoane-Sepúlveda, Juan B. (2010) The real plank problem and some applications. Proceedings of the American Mathematical Society, 138 (7). p. 2521. ISSN 0002-9939

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Official URL: http://www.ams.org/journals/proc/2010-138-07/S0002-9939-10-10295-0/


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Abstract

K. Ball has proved the "complex plank problem": if (x(k))(k=1)(n) is a sequence of norm I vectors in a complex Hilbert space (H, (., .)), then there exists a unit vector x for which |< x,x(k)>| >= 1/root n, k = 1,...,n. In general, this result is not true on real Hilbert spaces. However, in special cases we prove that the same result holds true. In general, for some unit vector x we have derived the estimate |< x,x(k)>| >= max{root lambda(1)/n, 1/root lambda(n)n}, where lambda(1) is the smallest and lambda(n) is the largest eigenvalue of the Hermitian matrix A = [(x(j), x(k))], j, k = 1,...,n. We have also improved known estimates for the norms of homogeneous polynomials which are products of linear forms on real Hilbert spaces.


Item Type:Article
Uncontrolled Keywords:Plank problems; Polarization constants; Product of linear functionals
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
ID Code:20003
Deposited On:20 Feb 2013 17:48
Last Modified:28 Nov 2016 09:28

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