On the Decomposition of an Operator into a Sum of Four Idempotents

We prove that operators of the form (2 ± 2/n)I + K are decomposable into a sum of four idempotents for integer n > 1 if there exists the decomposition K = K ₁ K ₂ ... K _n, , of a compact operator K. We show that the decomposition of the compact operator 4I + K or the operator K into a sum of four idempotents can exist if K is finite-dimensional. If n trK is a sufficiently large (or sufficiently small) integer and K is finite-dimensional, then the operator (2 – 2/n)I + K [or (2 + 2/n)I + K] is a sum of four idempotents.     

On the Decomposition of a Diagonal Operator into a Linear Combination of Idempotents or Projectors

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. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 3, pp. 388–393, March, 2005.