Proposition 3

Let E be a Banach space. Let $u$ be an endomorphism of E, and let A be its commutant in $\mathscr{L}(E)$. The closure B of $\pi (A)$ in the algebra $\mathscr{C}$alk(E) is a Banach algebra. In order that $u$ be a Riesz endomorphism of E, it is necessary and sufficient that $\pi (u)$ be invertible in B.

Since $\pi (A)$ is a unital subalgebra of $\mathscr{C}$alk(E), its closure B is a Banach algebra. If $u$ is a Riesz endomorphism of E, the element $\pi (u)$ possesses an inverse in $\pi (A)$ (cor. 1), hence in B. Conversely, suppose that $\pi (u)$ has an inverse $s$ in B. By definition of B, there exists an element $v$ of A such that

whence $\|1_E-\pi (u\circ v)\|<1$. According to prop. 2, the linear mapping $u\circ v$ is a Riesz endomorphism of E. The same is true of $u$ since $u$ and $v$ commute (III, p. 49, prop. 9).