I have been reading Kato's perturbation theory book and on spectral theory in general, and I have a question regarding spectral projections and their norms. I'll explain the general setup below.
$X$ is a complex Banach space and $\mathcal{L}:X\to X$ is a bounded operator. Let us suppose that there are two simple isolated eigenvalues $z_1$ and $z_2$ inside some circle $C$. We can define the spectral projection as $\pi=\frac{1}{2i\pi}\int_C (wI-\mathcal{L})^{-1}dw\in B(X)$, where $B(X)$ is the space of bounded operators on $X$.
Let us define similarly $\pi_1=\frac{1}{2i\pi}\int_{C_1} (wI-\mathcal{L})^{-1}dw\in B(X)$ to be the eigenprojection of $z_1$. Can we say that $\|\pi_1\|\le \|\pi\|?$ (operator norm)
Intuitively, it seems like it should hold (we are projecting onto a smaller region of the space). But I can not seem to show it. The best I can show is $\|\pi_1\|\le \|\pi_1\|\|\pi\|$ (since $\pi_1=\pi_1\circ \pi$). Does anyone know if this is true, or am I just goose hunting? Thanks!
This can fail already for finite dimensional Hilbert spaces. Let: $$A(e_2)=2 e_2, \qquad A(e_1+e_2)=e_1+e_2$$ ie $$A=\begin{pmatrix}1 & 0 \\ -1 & 2\end{pmatrix}.$$
The projection $\pi_1$ satisfies $\pi_1(e_1+e_2)=e_1+e_2$ and $\pi_1(e_2)=0$, hence it is equal to $\begin{pmatrix}1&1\\0&0\end{pmatrix}$. The projection $\pi$ is the identity, and clearly $\|\pi_1\|>1=\|\pi\|$.