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On decompositions of the identity operator into a linear combination of orthogonal projections

Abstract

In this paper we consider decompositions of the identity operator into a linear combination of $k\ge 5$ orthogonal projections with real coefficients. It is shown that if the sum $A$ of the coefficients is closed to an integer number between $2$ and $k-2$ then such a decomposition exists. If the coefficients are almost equal to each other, then the identity can be represented as a linear combination of orthogonal projections for $\frac{k-\sqrt{k^2-4k}}{2} < A < \frac{k+\sqrt{k^2-4k}}{2}$. In the case where some coefficients are sufficiently close to $1$ we find necessary conditions for the existence of the decomposition.

Article Information

 Title On decompositions of the identity operator into a linear combination of orthogonal projections Source Methods Funct. Anal. Topology, Vol. 16 (2010), no. 1, 57-68 MathSciNet MR2656132 Copyright The Author(s) 2010 (CC BY-SA)

Authors Information

S. Rabanovich
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Teresh\-chenkivs'ka, Kyiv, 01601, Ukraine

A. A. Yusenko
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Teresh\-chenkivs'ka, Kyiv, 01601, Ukraine

Citation Example

S. Rabanovich and A. A. Yusenko, On decompositions of the identity operator into a linear combination of orthogonal projections, Methods Funct. Anal. Topology 16 (2010), no. 1, 57-68.

BibTex

@article {MFAT543,
AUTHOR = {Rabanovich, S. and Yusenko, A. A.},
TITLE = {On decompositions of the identity operator into a linear combination of orthogonal projections},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {16},
YEAR = {2010},
NUMBER = {1},
PAGES = {57-68},
ISSN = {1029-3531},
URL = {http://mfat.imath.kiev.ua/article/?id=543},
}