Exercise 1, Appendix 4

Let K be a commutative field and E a vector space over K. We denote by End$^{pf}_K(E)$ the subset of End$_K(E)$ consisting of the endomorphisms of which a power belongs to End$^f_K(E)$. If $u\in$ End$^{pf}_K$ (E), then we set $I_u=\bigcap_{n\geqslant 0}$ Im $u^n$ and Tr $u=$ Tr($u|I_u$); this notion coincides with that defined in No. 2 for an endomorphism of finite rank. a) Let $u\in$ End$^{pf}_K(E)$; let V be a subspace of E stable under $u$, and let $u_V$ and $u_{E/V}$ be the endomorphisms of V and $E/V$ induced by $u$. Prove the equality Tr $u=$ Tr $u_V+$ Tr $u_{E/V}$.

b) Let U be a subspace of End$_K(E)$. Suppose that there exists an integer $n$ such that every product of $n$ elements of U has finite rank. Prove that the mapping Tr$: U\rightarrow K$ is linear.

c) Let F be a vector space over K, and let $u: E\rightarrow F$ and $v: F\rightarrow E$ be homomorphisms such that $u\circ v\in$ End$^{pf}_K(F)$. We have $v\circ u\in$ End$^{pf}_K(E)$ and Tr $v\circ u=$ Tr $u\circ v$.