On equivalence of linear combinations of idempotents

ABSTRACT

Let p and q be idempotent elements in a complex unital algebra and let $\mu ,\nu \in \mathbb{C}\setminus \left\{0\right\}$, $\xi \in \mathbb{C}$. In the present note we will prove that $\mu p+\nu q-\xi pq$ is equivalent to p+q (resp. p−q) if $\mu +\nu \ne \xi$ (resp. $\mu +\nu =\xi$). In particular, $\mu p+\nu q-\xi pq$ is invertible if and only if p+q (resp. p−q) is invertible, where $\mu +\nu \ne \xi$ (resp. $\mu +\nu =\xi$). We will also give alternative proofs of some already existing results.

COMMUNICATED BY:

• Ngai-Ching Wong

KEYWORDS:

• Idempotent
• linear combinations
• spectrum
• Fredholm

AMS SUBJECT CLASSIFICATIONS:

• 47A53
• 47A10

Disclosure statement

No potential conflict of interest was reported by the authors.