Definition 2

Let $u$ be a partial operator on E. One says that $u$ is normal if $u$ is closed with dense domain and if its bornification $b(u)$ is a normal endomorphism of E.

If $u\in \mathscr{L}$ (E), this definition agrees with EVT, V, p. 42, def. 4, by virtue of the formulas $1_E-b(u)^*b(u) = W(u) = (1_E+u^*u)^{-1}$ (prop. 1 of IV, p. 262) and $b(u^*) =b(u)^*$ (cor. of loc. cit.).

If $u$ is a self-adjoint partial operator on E, then $b(u)$ is hermitian (cor. of prop. 2 of IV, p. 264), therefore $u$ is normal.

Let D be the open unit disk in $\mathbf{C}$. We denote by $\beta$ the function from $\mathbf{C}$ into D defined by $\beta (z) =z/\surd\overline{1 + |z|^2}$. It is a homeomorphism, whose inverse satisfies $\beta^{-1}(z) =z/\surd\overline{1 - |z|^2}$ for $z\in D$.

Let $u\in \mathscr{L}(E)$. It follows from formulas (2) and (3) of proposition 1 of IV, p. 262 that $u=\beta^{-1}(b(u))$, and therefore