Proposition 13

Let E be a Fréchet space and $u\in \mathscr{L}(E)$.

a) The endomorphism $u$ is a Riesz endomorphism if and only if its nilspace N is finite-dimensional and its conilspace I is of finite codimension ;

b) Endow $E'$ with the topology of bounded convergence, the topology of compact convergence, or the weak topology. If $^tu$ is a Riesz endomorphism of $E'$, then $u$ is a Riesz endomorphism of E.

a) Suppose that the nilspace N is finite-dimensional and that the conilspace I is of finite codimension. Since Ker($u$)$\subset N$ and $I\subset$ Im($u$), the kernel of $u$ is finite-dimensional and the image of $u$ is of finite codimension. By definition and Proposition 11, the mapping $u$ is a Riesz endomorphism. The converse follows from the definition.

b) According to Proposition 12, the hypothesis implies that $u$ is a Fredholm endomorphism of index ind($^tu$) $=-$ ind($u$) $= 0$ (No.$^o3$, formula (4)). Let $n\in \mathbf{N}$. Since the image of $u^n$ is closed and has as orthogonal the kernel of $^tu^n$ (EVT, IV, p. 27, Prop. 2), the sequence (Im($u^n$)) is stationary, and $u$ is therefore a Riesz endomorphism.