On the Decomposition of a Diagonal Operator into a Linear Combination of Idempotents or Projectors - Rabanovych, V. I. (Ukrains’kyi Matematychnyi Zhurnal, 2005)
For a bounded operator that is not a sum of scalar and compact operators and is similar to a diagonal operator, we prove that it is a linear combination of three idempotents. It is also proved that any self-adjoint diagonal operator is a linear combination of four orthoprojectors with real coefficients.