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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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http://www.ams.org/ | Organisation |

## 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 |
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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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