2019
Том 71
№ 6

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

Rabanovych V. I.

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 1K 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.

English version (Springer): Ukrainian Mathematical Journal 56 (2004), no. 3, pp 512-519.

Citation Example: Rabanovych V. I. On the Decomposition of an Operator into a Sum of Four Idempotents // Ukr. Mat. Zh. - 2004. - 56, № 3. - pp. 419-424.

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