On decompositions of the identity operator into a linear combination of orthogonal projections
In this paper we consider decompositions of the identity operator into a linear combination of k5 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 k-k^2-4k 2 < A < k+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., number=1, journal=Methods of Functional Analysis and Topology, author=Yusenko, A. A. and Rabanovich, V. I., year=2010, month=Mar., pages=57–68