Corollary

Let $u$ be a self-adjoint partial operator on a complex Hilbert space E. If E is not zero, then the spectrum of $u$ is not empty.

Suppose that Sp($u$) is empty. Then $u$ is injective and $u^{-1}=-R(u,0)$ is an injective endomorphism of E such that Sp($u^{-1}$)$\subset \{0\}$ (prop. 15, a)), hence Sp($u^{-1}$) $=\{0\}($I, p. 26, cor. 1). Since $u^{-1}$ is normal (prop. 16), this implies that $u^{-1}$ is zero (I, p. 110, example 1), which is a contradiction.