Range of Riesz projection

Question

Is the range of Reisz projection of an isolated eigenvalue $\lambda$ of operator T over Banach space X equal to $\cup_{k\geq1}\ker(T-\lambda)^k$? It is not hard to show it when X is finite dimensional or when X is Hilbert and T self-adjoint or when resolvent admits a pole at $\lambda$. The problem arises when the resolvent admits essential singularity.

Answer 1

1. If $ \lambda$ is a pole of order $p$ of the resolvent of $T$, then the range of the Riesz projection is $\ker(T-\lambda)^p.$

2. If $\lambda$ is an essential singularity of the resolvent of $T$, then the range of the Riesz projection is

$$\{x \in X: \lim_{n \to \infty}||(T- \lambda)^n||^{1/n}=0 \}.$$

Reference: H:G. Heuser, Functional Analysis (Wiley & Sons), Chapters 49, 50.