On commutators of idempotents

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 $U=\left(\begin{array}{cc}{I}_1& 0\\ 0& -I_2\end{array}\right)$, where $I_i$ is the identity operator on the closed subspace $X_i$ ($i=1,2$). Furthermore, $T$ has necessarily the form $T=\left(\begin{array}{cc}0& \ast \\ \ast & 0\end{array}\right)$ with respect to the same decomposition. In this note we consider the question when $T$ is a commutator of the idempotent $P=\left(\begin{array}{cc}{I}_1& 0\\ 0& 0\end{array}\right)$ and some idempotent $Q$ on $X$. We also determine which scalar multiples of unilateral shifts on $l^p$ spaces ($1\le p\le \mathrm{\infty }$) are commutators of idempotent operators.