Functional Analysis, Mathematical Physics, and Dynamical Systems

On Decompositions of an Operator into a Sum of Projections

by Viacheslav Rabanovich (Institute of Mathematics NASU)

    Linear combinations of orthoprojections (in short, projections), in particular their sums, appear
in various problems of operator theory and its applications. A classical result is the spectral
theorem on decomposition of a self-adjoint operator with a finite spectrum into a linear
combination of projections onto the eigenspaces, the projections being pairwise orthogonal.
Excluding the orthogonality condition for projections leads to interesting. For instance, every
bounded self-adjoint operator is a linear combination of 4 projections (or an integral
combination of 5 projections). Also, operators from a wide class can be decomposed into sums
of a small number of projections. Such decompositions are used in different numerical
problems, especially where the calculation process can be parallelized, in information theory
(frames as codes), and in quantum information theory. We discuss the known facts on sums of
projections, describe general methods of constructing various decompositions, and partially
consider numerical applications.

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