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 k >= 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 k-root k2 -4k (2) < A < k+root k2 -4k 2. In the case where some coefficients are sufficiently close to 1 we find necessary conditions for the existence of the decomposition.