Some remarks on the mutually commutative idempotents

This paper deals mainly with the idempotency of an operator or a matrix T given by \(T=c_1 \Pi _1 +c_2 \Pi _2+\cdots +c_n\Pi _n,\) $T=c_1{\mathrm{\Pi }}_1+c_2{\mathrm{\Pi }}_2+\cdots +c_n{\mathrm{\Pi }}_n,$ where n is an arbitrary positive integer, \(\{\Pi _{1},\Pi _{2},\ldots ,\Pi _{n}\}\) $\{{\mathrm{\Pi }}_1,{\mathrm{\Pi }}_2,\dots ,{\mathrm{\Pi }}_n\}$ is a collection of mutually commutative idempotents, and \(c_1,c_2,\ldots ,c_n\) $c_1,c_2,\dots ,c_n$ are complex numbers. Some previous results in the cases of \(n=2\) $n=2$ and \(n=3\) $n=3$ are generalized, and meanwhile some new characterizations of the idempotency of T are obtained.