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Convex combinations of projections. (English) Zbl 0721.15007

The paper contains the following results: On an n-dimensional inner- product space, every operator T that satisfies \(0\leq T\leq 1\) is a convex combination of as few as \([\log_ 2n]+2\) projections and this number is sharp. If \(0\leq T\leq 1\) and trace T is a rational number, then T is an average of projections.
Further results are also got for the cases when the projections are needed to have the same rank / or to be commuting. In each case, the optimal number of projections is found out.
The convex combination part presented in this paper is very interesting.
Reviewer: S.Sridhar (Madras)

MSC:

15A30 Algebraic systems of matrices
15A63 Quadratic and bilinear forms, inner products
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
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References:

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