Lemma 11

Suppose that $E_b$ is finite-dimensional. Then, for every real number $\varepsilon >$ 0, the image of the spectral projector of $u$ defined by $[\varrho_e, \varrho_e+\varepsilon ]$ is infinite-dimensional.

If $E_b$ is finite-dimensional, then the essential spectrum of $u$ is nonempty, hence $\varrho_e$ is finite and belongs to Sp$_e(u)$. Moreover, the lower spectrum Sp$_b(u)$ is finite, hence there exists $\delta >0$ such that $[\varrho_e-\delta , \varrho_e+\delta ]\cap$ Sp($u$)$\subset [\varrho_e, \varrho_e+\delta ]$. The assertion then results from Lemma 9.