Exercise 7, § 7

Let $u$ be an endomorphism of a complex Hilbert space E such that the spectrum of $u$ is contained in the open unit disk of $\mathbf{C}$. Let F be the Hilbert space $\ell^2_E(\mathbf{N})$ of the sequences of square summable elements of E (EVT, V, p. 18, example) and let $s_F$ be the endomorphism of F such that

There exists a closed subspace H of F and a continuous bijective linear mapping $v: E\rightarrow H$ such that $u=v^{-1}s_Fv$.