Exercise 14, § 5

With the notation as in Ex. 13, suppose that $a \in S(E, F)$ is of order $\geq 1$. a) Show that for each $p \in \mathbf{N}$, $\sigma^{(p)}$ is of order $\geq p$. Deduce that the family $(\sigma^{(p)})_{p \in \mathbf{N}}$ is summable. Writing $A(\sigma) = \sum_p \sigma^{(p)}$, verify that for every $n \in \mathbf{N}$,

b) Show that if $\mathrm{Ker}(a) \supset \Gamma_m(E)$ for all $m \neq 1$, then $A(\sigma) = \Gamma(a \circ \gamma_1)$. c) Show that $A(\sigma)$ is a homomorphism of the cogebra $\Gamma(E)$ into the cogebra $\Gamma(F)$, compatible with the counits (cf. Ex. 13, e)). d) Show that if $a, \tau \in S(E, F)$ are two series of order $\geq 1$, then $A(\sigma + \tau) = A(\sigma) A(\tau)$, the product $A(\sigma) A(\tau)$ being defined by the product in $\Gamma(E)$ (cf. Ex. 13, c)).