Corollary

Let A be an open set in $\mathbf{C}$ and $n\in \mathbf{N}$. If A contains $n$ elements of Sp($u$), then the dimension of the image of the spectral projector $p_A$ of $u$ relative to A is at least equal to $n$.

Let $\lambda_1, . . .,\lambda_n$ be distinct elements of the spectrum of $u$ belonging to A. There exists a family $(V_i)_{1\leqslant i\leqslant n}$ of pairwise disjoint open sets of $\mathbf{C}$ such that $\lambda_i\in V_i$ for $1\leqslant i\leqslant n$. Let B be the union of the sets $V_i$. The image of $p_A$ contains the image of the spectral projector $p_B$; moreover $p_B$ is the sum of the projectors $p_{V_i}$, and since the image of $p_{V_i}$ is orthogonal to the image of $p_{V_j}$ for all $i\not =j$, the result follows from Prop. 10.