Lemma 2

Let E be a vector space and let $u$ be an endomorphism of E which possesses an index (A, V, p. 126).

If two of the following properties are satisfied, so is the third:

(i) The endomorphism $u$ is of index 0;

(ii) The nilespace N of $u$ is finite-dimensional;

(iii) The conilespace I of $u$ is of finite codimension.

For every integer $n\geqslant 0$, the endomorphism $u^n$ possesses an index, equal to $n$ ind($u$) (A, V, p. 127, Lemma 2). The sequence (dim(Ker($u^n$)))$_{n\in\mathbf{N}}$ of natural integers is increasing; for it to be stationary, it is necessary and sufficient that the nilespace of $u$ be finite-dimensional. The sequence (codim$_E$(Im($u^n$)))$_{n\in\mathbf{N}}$ of natural integers is increasing; for it to be stationary, it is necessary and sufficient that the conilespace of $u$ be of finite dimension. Suppose condition (i) satisfied; the relation

dim(Ker($u^n$))$-$ codim$_E$(Im($u^n$)) $=$ ind($u^n$) $=n$ ind($u$) $= 0$

then entails the equivalence of conditions (ii) and (iii). Conversely, if conditions (ii) and (iii) are satisfied, this formula entails that the sequence $(n$ ind($u$))$_{n\in\mathbf{N}}$ is stationary, hence $u$ is of index zero.