Products of Idempotents in Banach Algebras of Operators

Abstract

Let $X$ be a Banach space and $\mathcal{A}$ be the Banach algebra $B(X)$ of bounded (i.e. continuous) linear transformations (to be called operators) on $X$ to itself. Let $\mathcal{E}$ be the set of idempotents in $\mathcal{A}$ and $\mathcal{S}$ be the semigroup generated by $\mathcal{E}$ under composition as multiplication. If $T\in \mathcal{S}$ with $0\ne T\ne I_X$ then $T$ has a local block representation of the form $\left(\begin{array}{ccc}{T}_1& T_{2}0& 0\end{array}\right)$ on $X=Y\oplus Z$, a topological sum of non-zero closed subspaces $Y$ and $Z$ of $X$, and any $A\in \mathcal{A}$ has the form $\left(\begin{array}{ccc}{A}_1& A_2{A}_3& A_4\end{array}\right)$ with $T_1,A_1\in \mathcal{B}(Y)$, $T_2,A_2\in B(Z,Y),A_3\in B(Y,Z)$, and $A_4\in \mathcal{B}(Z)$. The purpose of this paper is to study conditions for $T$ to be in $\mathcal{S}$.

20 Pages