Galerkin approximation of holomorphic eigenvalue problems: weak T-coercivity and T-compatibility

1. Weak T-coercivity and T-compatibility

Let $X$ be a Hilbert space with scalar body $C$ and scalar product $⟨ \cdot, \cdot ⟩_{X}$ and associated norm $∥ \cdot ∥_{X}$ . Let $L (X)$ be the space of bounded linear operators from $X$ to $X$ with operator norm $∥ A ∥_{L (X)} := {sup}_{u \in X ∖ {0}} ∥ A u ∥_{X} / ∥ u ∥_{X}$ for $A \in L (X)$ . For $A \in L (X)$ we denote its adjoint operator by $A^{*} \in L (X)$ , i.e. $⟨ u, A^{*} v ⟩_{X} = ⟨ A u, v ⟩_{X}$ for all $u, v \in X$ . For a closed subspace $X_{n} \subset X$ let $L (X_{n})$ be the space of bounded linear operators from $X_{n}$ to $X_{n}$ with norm $∥ A_{n} ∥_{L (X_{n})} := {sup}_{u_{n} \in X_{n} ∖ {0}} ∥ A_{n} u_{n} ∥_{X} / ∥ u_{n} ∥_{X}$ for $A_{n} \in L (X_{n})$ and denote $P_{n}$ the orthogonal projection from $X$ to $X_{n}$ . Henceforth we assume that $(X_{n})_{n \in N}$ is a sequence of closed subspaces of $X$ such that $P_{n}$ converges point-wise to the identity, i.e. ${lim}_{n \in N} ∥ u - P_{n} u ∥_{X} = 0$ for each $u \in X$ .

Definition 1.1.

Let $A, T \in L (X)$ and $T$ be bijective. The operator $A$ is called

coercive, if ${inf}_{u \in X ∖ {0}} | ⟨ A u, u ⟩_{X} | / ∥ u ∥_{X}^{2} > 0$ ,
weakly coercive, if there exists a compact operator $K \in L (X)$ such that $A + K$ is coercive,
$T$ -coercive if $T^{*} A$ is coercive,
weakly $T$ -coercive if $T^{*} A$ is weakly coercive.

Due to the Lemma of Lax-Milgram every coercive operator is invertible. Every weakly $T$ -coercive operator is Fredholm with index zero. For a (weakly) coercive operator $A$ it is true that the Galerkin approximations $A_{n} = P_{n} A |_{X_{n}} \in L (X_{n})$ inherit the (weak) coercivity, while for (weakly) $T$ -coercive operators it is in general wrong.

We note that if $T^{*} A$ is weakly coercive, then $A T^{- 1}$ is so too. Vice-versa, if $A T$ is weakly coercive, then so is $T^{- *} A$ . Hence we could alternatively define $A$ to be (weakly) right $T$ -coercive, if $A T$ is (weakly) coercive. However, we stick to the former variant because it is more convenient.

For an operator $T \in L (X)$ or $T \in L (X_{n}),$ or a sum of such we define the “discrete norm”

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∥ T ∥_{n} := sup u_{n} \in X_{n} ∖ {0} \frac{∥ T u_{n} ∥_{X}}{∥ u_{n} ∥_{X}} = ∥ T ∥_{L (X_{n}, X)} = ∥ T P_{n} ∥_{L (X)} .

Definition 1.2.

Consider $T \in L (X)$ and $(T_{n} \in L (X_{n}))_{n \in N}$ . We say that $T_{n}$ converges to $T$ in discrete norm, if

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lim n \in N ∥ T - T_{n} ∥_{n} = 0.

We define in the following what we mean by $T$ -compatible approximations of weakly $T$ -coercive operators.

Definition 1.3.

Let $A \in L (X)$ be weakly $T$ -coercive. Then we call the sequence of Galerkin approximations $(A_{n} := P_{n} A |_{X_{n}} \in L (X_{n}))_{n \in N}$ $T$ -compatible, if $(A_{n})_{n \in N}$ is a sequence of index zero Fredholm operators and there exists a sequence of index zero Fredholm operators $(T_{n} \in L (X_{n}))_{n \in N}$ such that $T_{n}$ converges to $T$ in discrete norm: ${lim}_{n \in N} ∥ T - T_{n} ∥_{n} = 0$ .

Definition 1.4.

A sequence $(u_{n} \in X)_{n \in N}$ is said to be compact, if for every subsequence exists in turn a converging subsubsequence.

Definition 1.5.

A sequence $(A_{n} \in L (X_{n}))_{n \in N}$ is called regular, if for every bounded sequence $(u_{n} \in X_{n})_{n \in N}$ the compactness of $(A_{n} u_{n})_{n \in N}$ already implies the compactness of $(u_{n})_{n \in N}$ .

Next we briefly elaborate on the notion of regularity for readers who are totally unfamiliar with this concept. Regularity of Galerkin approximations is a meaningful generalization of stability and well suited for the approximation analysis of eigenvalue problems. Consider for example bijective $A \in L (X)$ and its Galerkin approximation $(A_{n} := P_{n} A |_{X_{n}} \in L (X_{n}))_{n \in N}$ . In this case regularity of $(A_{n})_{n \in N}$ implies stability: Assume that $(A_{n})_{n \in N}$ is not stable. Thus there exists $(u_{n} \in X_{n})_{n \in N}$ with $∥ u_{n} ∥_{X} = 1$ for each $n \in N$ such that ${lim}_{n \in N} ∥ A_{n} u_{n} ∥_{X} = 0$ . If $(A_{n})_{n \in N}$ is regular, there exists a subsequence $n (m)_{m \in N}$ and $u \in X$ such that ${lim}_{m \in N} u_{n (m)} = u$ . It follows $A u = {lim}_{m \in N} A_{n (m)} u_{n (m)} = 0$ . Since $A$ is bijective, it follows $u = 0$ which is a contradiction to $∥ u_{n (m)} ∥_{X} = 1$ .

On the other hand, consider a holomorphic Fredholm operator function $A (\cdot) : Λ \subset C \to L (X)$ with non-empty resolvent set and sequences $(λ_{n} \in Λ, u_{n} \in X_{n})_{n \in N}$ of eigenvalues with normalized eigenelements of the Galerkin approximation (i.e. $A_{n} (λ_{n}) u_{n} = 0$ ) such that ${lim}_{n \in N} λ_{n} = λ \in Λ$ (see Section 2 for definitions and details). If $A_{n} (λ)$ is regular for each $λ_{n} \in Λ$ , then $λ$ is indeed an eigenvalue of $A (\cdot)$ (i.e. there occurs no spectral pollution): Due to the continuity of $A_{n} (\cdot)$ with respect to $λ$ , $A_{n} (λ_{n}) u_{n} = 0$ implies ${lim}_{n \in N} A_{n} (λ) u_{n} = 0$ . If $(A_{n} (λ))_{n \in N}$ is regular, there exists a subsequence $n (m)_{m \in N}$ and $u \in X$ such that ${lim}_{m \in N} u_{n (m)} = u$ . It follows $A (λ) u = {lim}_{m \in N} A_{n (m)} (λ) u_{n (m)} = 0$ and $∥ u ∥_{X} = {lim}_{m \in N} ∥ u_{n (m)} ∥_{X} = 1$ , i.e. $λ$ is an eigenvalue of $A (\cdot)$ with normalized eigenelement $u$ .

Our next goal is to prove in Theorem 1.8 that $T$ -compatible Galerkin approximations of weakly $T$ -coercive operators are regular. In preparation we formulate the next two lemmata.

Lemma 1.6.

Let $T \in L (X) ∖ {0}$ and $(T_{n} \in L (X_{n}))_{n \in N}$ be a sequence of operators with $T_{n} \in L (X_{n})$ and ${lim}_{n \in N} ∥ T - T_{n} ∥_{n} = 0$ . Then there exist a constant $c > 0$ and an index $n_{0} \in N$ such that

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∥ T_{n} ∥_{L (X_{n})}, ∥ T_{n} ∥_{L (X_{n})}^{- 1} \leq c

for all $n > n_{0}$ . If $T$ is bijective and $T_{n}$ is Fredholm with index zero for each $n \in N$ , then there exist a constant $c > 0$ and an index $n_{0} \in N$ such that $T_{n}$ is also bijective for all $n > n_{0}$ and

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∥ (T_{n})^{- 1} ∥_{L (X_{n})} \leq c .

Proof.

Let $u_{n} \in X_{n}$ . With the triangle inequality we deduce

∥ T_{n} u_{n} ∥_{X} \leq ∥ T u_{n} ∥_{X} + ∥ (T - T_{n}) u_{n} ∥_{X}

and hence

∥ T_{n} ∥_{L (X_{n})}

\leq ∥ T ∥_{L (X)} + ∥ T - T_{n} ∥_{n} .

Since ${lim}_{n \in N} ∥ T - T_{n} ∥_{n} = 0$ the right hand side of the previous inequality is bounded. Similar, with the inverse triangle inequality we deduce

∥ T_{n} u_{n} ∥_{X} \geq ∥ T u_{n} ∥_{X} - ∥ (T - T_{n}) u_{n} ∥_{X}

and hence

∥ T_{n} ∥_{L (X_{n})}

\geq ∥ T ∥_{n} - ∥ T - T_{n} ∥_{n} .

It hold ${lim}_{n \in N} ∥ T ∥_{n} = ∥ T ∥_{L (X)} > 0$ and ${lim}_{n \in N} ∥ T - T_{n} ∥_{n} = 0$ . Thus let $n_{0} > 0$ be such that $| ∥ T ∥_{n} - ∥ T ∥_{L (X)} | < ∥ T ∥_{L (X)} / 3$ and $∥ T - T_{n} ∥_{n} < ∥ T ∥_{L (X)} / 3$ for all $n > n_{0}$ . It follows

∥ T_{n} ∥_{L (X_{n})}

\geq ∥ T ∥_{L (X)} / 3 > 0

for all $n > n_{0}$ . For the last claim let $n_{0} > 0$ be such that $∥ T - T_{n} ∥_{n} < \frac{1}{2 ∥ T^{- 1} ∥_{L (X)}}$ for all $n > n_{0}$ . Again with the inverse triangle inequality and

inf u \in X, ∥ u ∥_{X} = 1 ∥ T u ∥_{X} = 1 / ∥ T^{- 1} ∥_{L (X)} > 0

it follows

	$inf u_{n} \in X_{n}, ∥ u_{n} ∥_{X} = 1 ∥ T_{n} u_{n} ∥_{X}$	$\geq inf u \in X, ∥ u ∥_{X} = 1 ∥ T u ∥_{X} - ∥ T - T_{n} ∥_{n}$
		$\geq 1 / (2 ∥ T^{- 1} ∥_{L (X)})$

for all $n > n_{0}$ . We deduce that $T_{n}$ is injective. Since $T_{n}$ is Fredholm with index zero its bijectivity follows. The norm estimate holds due to ${inf}_{u_{n} \in X_{n}, ∥ u_{n} ∥_{X} = 1} ∥ T_{n} u_{n} ∥_{X} = 1 / ∥ T_{n}^{- 1} ∥_{L (X_{n})}$ . ∎

Lemma 1.7.

Let $A \in L (X)$ be weakly $T$ -coercive and $K \in L (X)$ be compact such that $T^{*} A + K$ is coercive. Let $(A_{n} := P_{n} A |_{X_{n}} \in L (X_{n}))_{n \in N}$ be a $T$ -compatible Galerkin approximation of $A$ . Then there exist $n_{0} \in N$ and $c > 0$ , such that $A_{n} + P_{n} T^{- *} K |_{X_{n}} \in L (X_{n})$ is invertible and

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∥ (A_{n} + P_{n} T^{- *} K |_{X_{n}})^{- 1} ∥_{L (X_{n})} \leq c

for all $n > n_{0}$ .

Proof.

Let $n$ be large enough such that $T_{n}$ is bijective (see Lemma 1.6). We compute

	$inf u_{n} \in X_{n} ∖ {0} sup v_{n} \in X_{n} ∖ {0}$	$\frac{\| ⟨ (A + T^{- *} K) u_{n}, v_{n} ⟩_{X} \|}{∥ u_{n} ∥_{X} ∥ v_{n} ∥_{X}}$
		$\geq inf u_{n} \in X_{n} ∖ {0} sup v_{n} \in X_{n} ∖ {0} \frac{\| ⟨ (A + T^{- *} K) u_{n}, T_{n} v_{n} ⟩_{X} \|}{∥ T_{n} ∥_{L (X_{n})} ∥ u_{n} ∥_{X} ∥ v_{n} ∥_{X}}$
		$\geq inf u_{n} \in X_{n} ∖ {0} sup v_{n} \in X_{n} ∖ {0} \frac{\| ⟨ ((A + T^{- *} K) u_{n}, T v_{n} ⟩_{X} \|}{∥ T_{n} ∥_{L (X_{n})} ∥ u_{n} ∥_{X} ∥ v_{n} ∥_{X}}$
		$- \frac{∥ A + T^{- *} K ∥_{L (X)}}{∥ T_{n} ∥_{L (X_{n})}} ∥ T - T_{n} ∥_{n}$
		$= inf u_{n} \in X_{n} ∖ {0} sup v_{n} \in X_{n} ∖ {0} \frac{\| ⟨ T^{} (A + T^{- } K) u_{n}, v_{n} ⟩_{X} \|}{∥ T_{n} ∥_{L (X_{n})} ∥ u_{n} ∥_{X} ∥ v_{n} ∥_{X}}$
		$- \frac{∥ A + T^{- *} K ∥_{L (X)}}{∥ T_{n} ∥_{L (X_{n})}} ∥ T - T_{n} ∥_{n}$
		$= inf u_{n} \in X_{n} ∖ {0} sup v_{n} \in X_{n} ∖ {0} \frac{\| ⟨ (T^{*} A + K) u_{n}, v_{n} ⟩_{X} \|}{∥ T_{n} ∥_{L (X_{n})} ∥ u_{n} ∥_{X} ∥ v_{n} ∥_{X}}$
		$- \frac{∥ A + T^{- *} K ∥_{L (X)}}{∥ T_{n} ∥_{L (X_{n})}} ∥ T - T_{n} ∥_{n}$
		$\geq ~ c ∥ T_{n} ∥_{L (X_{n})}^{- 1} - \frac{∥ A + T^{- *} K ∥_{L (X)}}{∥ T_{n} ∥_{L (X_{n})}} ∥ T - T_{n} ∥_{n}$

with coercivity constant

~ c := inf u \in X ∖ {0} | ⟨ (T^{*} A + K) u, u ⟩_{X} | / ∥ u ∥_{X}^{2} > 0.

Since $∥ T_{n} ∥_{L (X_{n})}$ is uniformly bounded from above and below (see Lemma 1.6) and $T_{n}$ converges to $T$ in discrete norm by assumption, it follows the existence of $n_{0} \in N$ and $c > 0$ such that

inf u_{n} \in X_{n} ∖ {0} sup v_{n} \in X_{n} ∖ {0} \frac{| ⟨ (A + T^{- *} K) u_{n}, v_{n} ⟩_{X} |}{∥ u_{n} ∥_{X} ∥ v_{n} ∥_{X}} \geq c

for all $n > n_{0}$ . Hence $A_{n} + P_{n} T^{- *} K |_{X_{n}}$ is injective. Since $A_{n}$ is Fredholm with index zero and $K$ is compact, $A_{n} + P_{n} T^{- *} K |_{X_{n}}$ is Fredholm with index zero too. Thus $A_{n} + P_{n} T^{- *} K |_{X_{n}}$ is bijective. The norm estimate follows now from

	$inf u_{n} \in X_{n} ∖ {0} sup v_{n} \in X_{n} ∖ {0} \frac{\| ⟨ B_{n} u_{n}, v_{n} ⟩_{X} \|}{∥ u_{n} ∥_{X} ∥ v_{n} ∥_{X}}$	$= inf u_{n} \in X_{n} ∖ {0} \frac{∥ B_{n} u_{n} ∥_{X}}{∥ u_{n} ∥_{X}}$
		$= {(sup u_{n} \in X_{n} ∖ {0} \frac{∥ u_{n} ∥_{X}}{∥ B_{n} u_{n} ∥_{X}})}^{- 1}$
		$= ∥ B_{n}^{- 1} ∥_{L (X_{n})}^{- 1}$

for any bijective $B_{n} \in L (X_{n})$ . ∎

Theorem 1.8.

Let $A \in L (X)$ be weakly $T$ -coercive and

(A_{n} := P_{n} A |_{X_{n}} \in L (X_{n}))_{n \in N}

be a $T$ -compatible Galerkin approximation. Then $(A_{n})_{n \in N}$ is regular.

Proof.

Without loss of generality let $(u_{n} \in L (X_{n}))_{n \in N}$ be a bounded sequence, $(A_{n} u_{n})_{n \in N}$ and $u^{'} \in X$ be such that ${lim}_{n \in N} A_{n} u_{n} = u^{'}$ . Let $K \in L (X)$ be compact such that $T^{*} A + K$ is coercive. Let $~ A := A + T^{- *} K$ and ${~ A}_{n} := P_{n} ~ A |_{X_{n}}$ . Since $K$ is compact and $(u_{n})_{n \in N}$ is bounded, there exist a subsequence $(u_{n (m)})_{m \in N}$ and $u^{''} \in X$ such that ${lim}_{m \in N} T^{- *} K u_{n (m)} = u^{''}$ . It follows

lim m \in N {~ A}_{n (m)} u_{n (m)} = u^{'} + u^{''} .

Due to Lemma 1.7 there exist $c > 0$ and $m_{0} \in N$ , such that for all $m > m_{0}$ operator ${~ A}_{n (m)}$ is invertible and $∥ {~ A}_{n (m)}^{- 1} ∥_{L (X_{n (m)})} \leq c$ . For $m > m_{0}$ we compute

	$∥ u_{n (m)} -$	${~ A}^{- 1} (u^{'} + u^{''}) ∥_{X}$
		$\leq ∥ u_{n (m)} - P_{n (m)} {~ A}^{- 1} (u^{'} + u^{''}) ∥_{X} + ∥ (I - P_{n (m)}) {~ A}^{- 1} (u^{'} + u^{''}) ∥_{X}$
		$\leq c ∥ {~ A}_{n (m)} u_{n (m)} - {~ A}_{n (m)} P_{n (m)} {~ A}^{- 1} (u^{'} + u^{''}) ∥_{X}$
		$+ ∥ (I - P_{n (m)}) {~ A}^{- 1} (u^{'} + u^{''}) ∥_{X}$
		$\leq c ∥ {~ A}_{n (m)} u_{n (m)} - (u^{'} + u^{''}) ∥_{X}$
		$+ c ∥ (I - {~ A}_{n (m)} P_{n (m)} {~ A}^{- 1}) (u^{'} + u^{''}) ∥_{X}$
		$+ ∥ (I - P_{n (m)}) {~ A}^{- 1} (u^{'} + u^{''}) ∥_{X} .$

The first term on the right hand side of the latter inequality converges to zero, as previously discussed. The second and third term converge to zero, because $(P_{n (m)})_{m \in N}$ converges point-wise to the identity. Hence

lim m \in N u_{n (m)} = {~ A}^{- 1} (u^{'} + u^{''}) .

∎

2. Holomorphic eigenvalue problems

We refer the reader to [13] and [20, Appendix] for theory on holomorphic (Fredholm) operator functions. Let $Λ \subset C$ be an open, connected and non-empty subset of $C$ . Let $A (\cdot) : Λ \to L (X)$ be an operator function. An operator function $A (\cdot)$ is called holomorphic, if it is complex differentiable. An operator function $A (\cdot)$ is called Fredholm, if $A (λ)$ is Fredholm for each $λ \in Λ$ . We denote the resolvent set and spectrum of an operator function $A (\cdot) : Λ \to L (X)$ as

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ρ (A (\cdot)) := {λ \in Λ : A (λ) is invertible} and σ (A (\cdot)) := Λ ∖ ρ (A (\cdot)) .

For an operator function $A (\cdot) : Λ \to L (X)$ we denote by $A^{*} (\cdot)$ the operator function defined by $A^{*} (λ) := A (λ)^{*}$ for each $λ \in Λ$ and by $A^{- 1} (\cdot) : ρ (A (\cdot)) \to L (X)$ the operator function defined by $A^{- 1} (λ) := A (λ)^{- 1}$ for each $λ \in ρ (A (\cdot))$ . Note that for a holomorphic operator function $A (\cdot) : Λ \to L (X)$ the operator function defined by $λ \mapsto A^{*} (¯ ¯ ¯ λ)$ is holomorphic as well. For a holomorphic operator function $A (\cdot) : Λ \to L (X)$ denote by $A^{(j)} (\cdot) : Λ \to L (X)$ the $j^{t h}$ derivative of $A (\cdot) : Λ \to L (X)$ . It is well known (see e.g. [12, Theorem 8.2]) that for a holomorphic Fredholm operator function $A (\cdot) : Λ \to L (X)$ such that $A (λ)$ is bijective for at least one $λ \in Λ$ , the spectrum $σ (A (\cdot))$ is discrete, has no accumulation points in $Λ$ and every $λ \in σ (A (\cdot))$ is an eigenvalue. That is, there exists $u \in X$ such that $A (λ) u = 0$ . In this case we call $u$ an eigenelement. An ordered collection of elements $(u_{0}, u_{1}, \dots, u_{m - 1})$ in $X$ is called a Jordan chain at $λ$ if $u_{0}$ is an eigenelement corresponding to $λ$ and if

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l \sum j = 0 \frac{1}{j!} A^{(j)} (λ) u_{l - j} for l = 0, 1, \dots, m - 1.

The elements of a Jordan chain are called generalized eigenelements and the closed linear hull of all generalized eigenelements of $A (\cdot)$ at $λ$ is called the generalized eigenspace $G (A (\cdot), λ)$ for $A (\cdot)$ at $λ$ . For an eigenelement $u \in ker A (λ) ∖ {0}$ we denote by $ϰ (A (\cdot), λ, u)$ the maximal length of a Jordan chain at $λ$ beginning with $u$ and

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ϰ (A (\cdot), λ) := max u \in ker A (λ) ∖ {0} ϰ (A (\cdot), λ, u) .

The maximal length of a Jordan chain $ϰ (A (\cdot), λ)$ is always finite, see e.g. [20, Lemma A.8.3]. Next we generalize Definitions 1.1, 1.3, 1.5 and Theorem 1.8 to operator functions.

Definition 2.1.

Let $A (\cdot), T (\cdot) : Λ \to L (X)$ be operator functions and $ρ (T (\cdot)) = Λ$ . $A (\cdot)$ is (weakly) ( $T (\cdot)$ -)coercive, if $A (λ)$ is (weakly) ( $T (λ)$ -)coercive for each $λ \in Λ$ .

Definition 2.2.

Let $A (\cdot) : Λ \to L (X)$ be weakly $T (\cdot)$ -coercive. Then we call the sequence of Galerkin approximations $(A_{n} (\cdot) := P_{n} A (\cdot) |_{X_{n}} : Λ \to L (X_{n}))_{n \in N}$ $T (\cdot)$ -compatible, if $(A_{n} (λ))_{n \in N}$ is $T (λ)$ compatible for each $λ \in Λ$ .

Definition 2.3.

Let $A (\cdot) : Λ \to L (X)$ be an operator function. The sequence of Galerkin approximations $(A_{n} (\cdot) := P_{n} A (\cdot) |_{X_{n}} : Λ \to L (X_{n}))_{n \in N}$ is regular, if $(A_{n} (λ))_{n \in N}$ is regular for each $λ \in Λ$

Theorem 2.4.

Let $A (\cdot) : Λ \to L (X)$ be weakly $T (\cdot)$ -coercive and

(A_{n} (\cdot) := P_{n} A (\cdot) |_{X_{n}} : Λ \to L (X_{n}))_{n \in N}

be a $T (\cdot)$ -compatible Galerkin approximation. Then $(A_{n} (\cdot))_{n \in N}$ is regular.

Proof.

Follows from Theorem 2.4. ∎

Next we prepare to apply [18], [19].

Lemma 2.5.

Let $A (\cdot) : Λ \to L (X)$ be a holomorphic Fredholm operator function and let $(X_{n})_{n \in N}$ be a sequence of closed subspaces of $X$ with orthogonal projections $P_{n}$ onto $X_{n}$ , such that $(P_{n})_{n \in N}$ converges point-wise to the identity. Then the Galerkin scheme $(P_{n} A (\cdot) |_{X_{n}})_{n \in N}$ is a discrete approximation scheme in the sense of [18].

Proof.

For a Galerkin scheme it holds with the notation of [18]

U

= V = X, X_{n} = Y_{n} = X_{n}, A_{n} (\cdot) = P_{n} A (\cdot) |_{X_{n}}, p_{n} = q_{n} = P_{n} .

Assumptions a1)-a4) of [18] follow all from the point-wise convergence of $P_{n}$ . ∎

Next we generalize Theorem 4.3.7 of [22].

Lemma 2.6.

Let $Λ \subset C$ be open, $X$ be a Hilbert space and $L (X)$ be the space of bounded linear operators from $X$ to $X$ . Let $A (\cdot) : Λ \to L (X)$ be a holomorphic Fredholm operator function with non-empty resolvent set and $(X_{n})_{n \in N}$ be a sequence of closed subspaces of $X$ with orthogonal projections $P_{n}$ onto $X_{n}$ , such that $(P_{n})_{n \in N}$ converges point-wise to the identity, i.e. ${lim}_{n \in N} ∥ u - P_{n} u ∥_{X} = 0$ for all $u \in X$ . Let $A_{n} (\cdot) : Λ \to L (X_{n})$ be the Galerkin approximation of $A (\cdot)$ defined by $A_{n} (λ) := P_{n} A (λ) |_{X_{n}}$ for each $λ \in Λ$ . Let the assumptions of [18, Theorem 2, Theorem 3] and [19, Theorem 2, Theorem 3] be satisfied. Let $~ Λ \subset Λ$ be a compact set with rectifiable boundary $\partial ~ Λ \subset ρ (A (\cdot))$ and $~ Λ \cap σ (A (\cdot)) = {λ_{0}}$ . Then there exist $n_{0} \in N$ and $c > 0$ such that for all $n > n_{0}$

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for all $λ_{n} \in σ (A_{n} (\cdot)) \cap ~ Λ$ and all $u_{n} \in ker A_{n} (λ_{n})$ with $∥ u_{n} ∥_{X} = 1$ .

Proof.

We proceed as in [22]: Theorem 4.3.7 of [22] requires a special form of the operator function $A (\cdot)$ . However its proof uses this assumption only to apply Lemma 4.2.1 of [22]. Hence we need to establish the result of [22, Lemma 4.2.1] without the assumption on the special form of $A (\cdot)$ . However, the result of [22, Lemma 4.2.1] already follows from [18, Theorem 2 ii)].

∎

Next we apply [18], [19] and Lemma 2.6.

Proposition 2.7.

Let $Λ \subset C$ be open, connected and non-empty, $X$ be a Hilbert space and $L (X)$ be the space of bounded linear operators from $X$ to itself. Let $A (\cdot) : Λ \to L (X)$ be a holomorphic Fredholm operator function with non-empty resolvent set $ρ (A (\cdot)) \neq \emptyset$ . Let $(X_{n})_{n \in N}$ be a sequence of closed subspaces of $X$ with orthogonal projections $P_{n}$ onto $X_{n}$ , such that $(P_{n})_{n \in N}$ converges point-wise to the identity, i.e. ${lim}_{n \in N} ∥ u - P_{n} u ∥_{X} = 0$ for each $u \in X$ . Let $A_{n} (\cdot) : Λ \to L (X_{n})$ be the Galerkin approximation of $A (\cdot)$ defined by $A_{n} (λ) := P_{n} A (λ) |_{X_{n}}$ for each $λ \in Λ$ . Assume that $A_{n} (λ)$ is Fredholm with index zero for each $λ \in Λ$ and $n \in N$ . Assume that $(A_{n} (\cdot))_{n \in N}$ is a regular approximation of $A (\cdot)$ (see Definition 2.3). Then the following results hold.

For every eigenvalue $λ_{0}$ of $A (\cdot)$ exists a sequence $(λ_{n})_{n \in N}$ converging to $λ_{0}$ with $λ_{n}$ being an eigenvalue of $A_{n} (\cdot)$ for almost all $n \in N$ .
Let $(λ_{n}, u_{n})_{n \in N}$ be a sequence of normalized eigenpairs of $A_{n} (\cdot)$ , i.e.

$A_{n} (λ_{n}) u_{n} = 0,$

and $∥ u_{n} ∥_{X} = 1$ , so that $λ_{n} \to λ_{0} \in Λ$ , then
1. $λ_{0}$ is an eigenvalue of $A (\cdot)$ ,
2. $(u_{n})_{n \in N}$ is a compact sequence and its cluster points are normalized eigenelements of $A (λ_{0})$ .
For every compact $~ Λ \subset ρ (A)$ the sequence $(A_{n} (\cdot))_{n \in N}$ is stable on $~ Λ$ , i.e. there exist $n_{0} \in N$ and $c > 0$ such that $∥ A_{n} (λ)^{- 1} ∥_{L (X_{n})} \leq c$ for all $n > n_{0}$ and all $λ \in ~ Λ$ .
For every compact $~ Λ \subset Λ$ with rectifiable boundary $\partial ~ Λ \subset ρ (A (\cdot))$ exists an index $n_{0} \in N$ such that

(12) $dim G (A (\cdot), λ_{0}) = \sum λ_{n} \in σ (A_{n} (\cdot)) \cap ~ Λ dim G (A_{n} (\cdot), λ_{n}) .$

for all $n > n_{0}$ , whereby $G (B (\cdot), λ)$ denotes the generalized eigenspace of an operator function $B (\cdot)$ at $λ \in Λ$ .

Let $~ Λ \subset Λ$ be a compact set with rectifiable boundary $\partial ~ Λ \subset ρ (A (\cdot))$ , $~ Λ \cap σ (A (\cdot)) = {λ_{0}}$ and

(13)

\begin{matrix} δ_{n} & := max \begin{matrix} u_{0} \in G (A (\cdot), λ_{0}) ∥ u_{0} ∥_{X} \leq 1 \end{matrix} inf u_{n} \in X_{n} ∥ u_{0} - u_{n} ∥_{X}, δ_{n}^{*} & := max \begin{matrix} u_{0} \in G (A^{*} (¯ \cdot), λ_{0}) ∥ u_{0} ∥_{X} \leq 1 \end{matrix} inf u_{n} \in X_{n} ∥ u_{0} - u_{n} ∥_{X}, \end{matrix}

whereby $_{0}$ denotes the complex conjugate of $λ_{0}$ and $A^{*} (\cdot)$ the adjoint operator function of $A (\cdot)$ defined by $A^{*} (λ) := A (λ)^{*}$ for each $λ \in Λ$ . Then there exist $n \in N$ and $c > 0$ such that for all $n > n_{0}$

(14) $| λ_{0} - λ_{n} | \leq c (δ_{n} δ_{n}^{*})^{1 / ϰ (A (\cdot), λ_{0})}$

for all $λ_{n} \in σ (A_{n} (\cdot)) \cap ~ Λ$ , whereby $ϰ (A (\cdot), λ_{0})$ denotes the maximal length of a Jordan chain of $A (\cdot)$ at the eigenvalue $λ_{0}$ ,
(15) $| λ_{0} - {¯ λ}_{n} | \leq c δ_{n} δ_{n}^{*}$

whereby ${¯ λ}_{n}$ is the weighted mean of all the eigenvalues of $A_{n} (\cdot)$ in $~ Λ$

(16)
(17) $\begin{matrix} inf u_{0} \in ker A (λ_{0}) ∥ u_{n} - u_{0} ∥_{X} & \leq c (| λ_{n} - λ_{0} | + max \begin{matrix} u_{0}^{'} \in ker A (λ_{0}) ∥ u_{0}^{'} ∥_{X} \leq 1 \end{matrix} inf u_{n}^{'} \in X_{n} ∥ u_{0}^{'} - u_{n}^{'} ∥_{X}) \leq c (c (δ_{n} δ_{n}^{*})^{1 / ϰ (A (\cdot), λ_{0})} + δ_{n}) \end{matrix}$

for all $λ_{n} \in σ (A_{n} (\cdot)) \cap ~ Λ$ and all $u_{n} \in ker A_{n} (λ_{n})$ with $∥ u_{n} ∥_{X} = 1$ .

Proof.

The first three claims follow with [18, Theorem 2], if we can proof that the required assumptions are satisfied. First of all a Galerkin scheme is a discrete approximation scheme due to Lemma 2.5. The operator function $A (\cdot)$ are holomorphic by assumption. It follows that $A_{n} (\cdot) := P_{n} A (\cdot) P_{n} |_{X_{n}}$ is also holomorphic. $A (\cdot)$ and $A_{n} (\cdot)$ are index zero Fredholm operator functions by assumption. Assumption b1 $ρ (A (\cdot)$

Galerkin approximation of holomorphic eigenvalue problems: weak T-coercivity and T-compatibility

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1. Weak T-coercivity and T-compatibility

Definition 1.1.

Definition 1.2.

Definition 1.3.

Definition 1.4.

Definition 1.5.

Lemma 1.6.

Proof.

Lemma 1.7.

Proof.

Theorem 1.8.

Proof.

2. Holomorphic eigenvalue problems

Definition 2.1.

Definition 2.2.

Definition 2.3.

Theorem 2.4.

Proof.

Lemma 2.5.

Proof.

Lemma 2.6.

Proof.

Proposition 2.7.

Proof.

Galerkin approximation of holomorphic eigenvalue problems: weak T-coercivity and T-compatibility

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This week in AI

1. Weak T-coercivity and T-compatibility

Definition 1.1.

Definition 1.2.

Definition 1.3.

Definition 1.4.

Definition 1.5.

Lemma 1.6.

Proof.

Lemma 1.7.

Proof.

Theorem 1.8.

Proof.

2. Holomorphic eigenvalue problems

Definition 2.1.

Definition 2.2.

Definition 2.3.

Theorem 2.4.

Proof.

Lemma 2.5.

Proof.

Lemma 2.6.

Proof.

Proposition 2.7.

Proof.