On commutators of idempotents

Abstract: Let $T$ be an operator on Banach space $X$ that is similar to $- T$ via an involution $U$. Then $U$ decomposes the Banach space $X$ as $X = X_1 \oplus X_2$ with respect to which decomposition we have $U = \left(\begin{matrix} I_1 & 0 \ 0 & -I_2 \end{matrix} \right)$, where $I_i$ is the identity operator on the closed subspace $X_i$ ($i=1, 2$). Furthermore, $T$ has necessarily the form $T = \left(\begin{matrix} 0 & * \ * & 0 \end{matrix} \right) $ with respect to the same decomposition. In this note we consider the question when $T$ is a commutator of the idempotent $P = \left(\begin{matrix} I_1 & 0 \ 0 & 0 \end{matrix} \right)$ and some idempotent $Q$ on $X$. We also determine which scalar multiples of unilateral shifts on $l^p$ spaces ($1 \le p \le \infty$) are commutators of idempotent operators.