Doubt in proof of $\lim_{n \to \infty} [\lambda R(\lambda, A)]^n = P.$

Question

I have a doubt in proof of Lemma $4.7$ of this paper.

---

Lemma: Let $A$ be a closed operator on a complex Banach space $E$ and assume that $0$ is an eigenvalue of $A$ and a pole of the resolvent $R(\cdot, A).$ Denote by $P$ the corresponding spectral projection.

$(i)$ If $\lambda > 0$ is contained in $\rho(A)$ and if the operator family $([\lambda R(\lambda, A)]^n)_{n \in \mathbb N}$ is bounded then $0$ is a simple pole of the resolvent.

$(ii)$ Suppose in addition that $0=s(A).$ If $\lambda >0$ and $s(A)=0$ is a simple pole of the resolvent, then $$\lim_{n \to \infty} [\lambda R(\lambda, A)]^n = P.$$

---

In the proof of $(ii),$ the author first shows that there exists $c \in (0,1)$ such that $|\mu| \leq c$ for all $\mu \in \sigma(\lambda R(\lambda,A)) \setminus \{1\}.$ He then says, in particular $$r(\lambda R(\lambda,A)(I-P))\leq c<1.$$

---

I don't understand how the last statement follows in particular. Is there any reason why $$\sigma(\lambda R(\lambda,A)(I-P))\subseteq \sigma(\lambda R(\lambda,A))?$$

Answer 1

Here are a few details which might be helpful to understand the argument in the paper better:

(1) First note that $P$ is also the spectral projection for the spectral value $\frac{1}{\lambda}$ of the operator $R(\lambda,A)$; this result can, for instance, be found in [Engel/Nagel: One-Parameter Semigroups for Linear Evolution Equations (2000), Proposition IV.1.18]. From this, one easily deduces that $P$ is the spectral projection for the spectral value $1$ of the operator $\lambda R(\lambda,A)$.

(2) Now one uses the following general result about spectral projections (this result is essentially the reason why spectral projections are useful): if $T$ is a bounded linear operator on a complex Banach space (in our case, $T = \lambda R(\lambda,A)$) and if the spectrum $\sigma(T)$ can be decomposed as $\sigma(T) = \sigma_0 \cup \sigma_1$, where $\sigma_0$ and $\sigma_1$ are non-empty, closed and disjoint, then the spectral projection $P$ associated to $\sigma_0$ has the following properties:

• $P$ commutes with $T$, so both the range of $P$ and the range of the complementary projection $(I-P)$ are invariant under $T$.

• The spectrum of the restricted linear operator $T|_{PE}$ on $PE$ coincides with $\sigma_0$ and the spectrum of $T|_{(I-P)E}$ coincides with $\sigma_1$.

  In particular, it follows that $\sigma(T(I-P)) = \sigma_1 \cup \{0\}$ since $E = PE \oplus (I-P)E$ and since $T(I-P)$ acts as $T|_{(I-P)E}$ on $(I-P)E$ and as the zero operator on $PE$.

[Remark: Similar results are also true if $T$ is not bounded but only closed - but in this case one has to assume that $\sigma_0$ is compact.]

References:

• Details about the above mentioned decomposition can, for instance, be found in [Engel/Nagel (2000), Proposition IV.1.16] (although the term "spectral projection" is not explicitly used there).

• I also recommend the spectral theory chapters in Yosida's "Functional Analysis", Kato's "Perturbation Theory for Linear Operators" and in Taylor's "Introduction to Functional Analysis" (yet, I do not have those books at hand right now, so I cannot give you precise references).

• One some occasions I had the impression that some spectral theoretical results which are often employed in the theory of $C_0$-semigroups were scattered over several references, which makes it difficult to quote them in a concise way. Therefore, I wrote a short appendix for my PhD thesis where I summarized some results from spectral theory in a way that makes them very easy to use in the theory of $C_0$-semigroups; you can find the thesis under the following DOI: 10.18725/OPARU-4238. In the context of the question, the appendices A.1 and A.3 are probably most interesting (and maybe also A.2). The result about spectral projections quoted above can be found (in a more general version) in Theorem A.1.2. Note that the appendix contains almost no proofs; it is rather a summary of results from various places in the literature.