I have a doubt in proof of Lemma $4.7$ of [this paper.](https://arxiv.org/pdf/1511.09020.pdf)




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**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
$$\lim\_{n \to \infty} [\lambda R(\lambda, A)]^n = P.$$




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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 $$r(\lambda R(\lambda,A)(I-P))\leq c<1.$$




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



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


