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

Abstract

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, \(\sum\nolimits_1^n {K_i = 0} \) , 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 tr K 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.