# Spectral theory; separation of spectrum, eigenprojections

I have been reading Kato's perturbation theory book and on spectral theory in general, and I have a question regarding the separation of spectrum (c.f. Kato, Perturbation theory for linear operators, Theorem III.6.18). I'll explain the general setup below.

$$X$$ is a complex Banach space and $$\mathcal{L}:X\to X$$ is a bounded operator. For $$z\in\sigma(\mathcal{L})$$ which is isolated, he defines the spectral projection as $$\pi_z=\frac{1}{2i\pi}\int_C (wI-\mathcal{L})^{-1}dw\in B(X)$$, where $$C$$ is a small circle around $$z$$, and $$B(X)$$ is the space of bounded operators on $$X$$.

When $$\mathcal{L}$$ is finite rank, he mentions that $$\pi_z$$ is the projection onto the generalized eigenspace associated to $$z$$.

Does this naturally extend to operators which are not finite rank; for example if $$z$$ is an isolated eigenvalue such that $$\pi_z$$ has finite dimensional range, then can we say $$\pi_z$$ projects onto the generalised eigenspace of $$z$$?

• Yes, exactly. You can find that in the book of Dunford and Schwarz ("Linear Operators, Part I"). The assumption on the finite-dimensional range of the projection is crucial here and cannot be dropped. – amsmath Aug 30 '19 at 20:45
• Thanks for the reference! :) – Artur Aug 30 '19 at 20:53
• You're welcome. – amsmath Aug 30 '19 at 21:06

If $$z$$ is an isolated point of the spectrum, then $$\pi_z$$ is well-defined by the integral around $$z$$ that encloses no other points of the spectrum. And, $$(\mathcal{L}-zI)\pi_z = \frac{1}{2\pi i}\oint_{C}(\mathcal{L}-zI)(wI-\mathcal{L})^{-1}dw \\ = \frac{1}{2\pi i}\oint_{C}(w-z)(wI-\mathcal{L})^{-1}dw$$ The same is true for all positive powers of $$\mathcal{L}-zI$$, i.e., $$(\mathcal{L}-zI)^n$$ is turned into $$(w-z)^{n}$$ inside the integral for $$n=1,2,3,\cdots$$: $$(\mathcal{L}-zI)^n\pi_z =\frac{1}{2\pi i}\oint_{C}(w-z)^{n}(wI-\mathcal{L})^{-1}dw.$$ These give the negative coefficients in the Laurent series expansion of the resolvent around $$z$$. And $$(\mathcal{L}-zI)$$ implements a ladder from one coefficient to the next. $$(\mathcal{L}-zI)^n\pi_z$$ is eventually found to be $$0$$ by making assumptions about rank.