Products of Idempotents in Banach Algebras of Operators

TL;DR

This paper investigates the structure and conditions under which products of idempotent operators in the algebra of bounded linear operators on a Banach space can be characterized, focusing on their local block representations.

Contribution

It provides a detailed analysis of the local block form of operators in the semigroup generated by idempotents and establishes conditions for such operators to belong to this semigroup.

Findings

Operators have a specific local block form on a decomposition of the space.

Conditions are identified for operators to be in the semigroup generated by idempotents.

The structure of these operators relates to their block components and subspace decompositions.

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 $\begin{pmatrix} T_1 & T_2 0 & 0 \end{pmatrix}$ 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 $\begin{pmatrix} A_1 & A_2 A_3 & A_4 \end{pmatrix}$ 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 $\mathcal{S}$.