Local Convergence of Spectra and Pseudospectra

J. Spectr. Theory 8 (2018), 1051–1098 Journal of Spectral Theory DOI 10.4171/JST/222 © European Mathematical Society

Local convergence of spectra and pseudospectra

Sabine Bögli

Abstract. We prove local convergence results for the spectra and pseudospectra of sequences of linear operators acting in diﬀerent Hilbert spaces and converging in generalised strong resolvent sense to an operator with possibly non-empty essential spectrum. We establish local spectral exactness outside the limiting essential spectrum, local "- pseudospectral exactness outside the limiting essential "-near spectrum, and discuss properties of these two notions including perturbation results.

Mathematics Subject Classiﬁcation (2010). 47A10, 47A55, 47A58.

Keywords. Eigenvalue approximation, spectral exactness, spectral inclusion, spectral pollution, resolvent convergence, pseudospectra.

Contents

1 Introduction...... 1051 2 Localconvergenceofspectra...... 1054 3 Localconvergenceofpseudospectra ...... 1067 4 ApplicationsandExamples...... 1080 References...... 1095

1. Introduction

We address the problem of convergence of spectra and pseudospectra for a sequence .Tn/n2N of closed linear operators approximating an operator T . We establish regions K C of local convergence,

lim .Tn/ K .T/ K; (1.1) n!1 \ D \

lim ".Tn/ K ".T/ K; " > 0; (1.2) n!1 \ D \ 1052 S. Bögli where the limits are deﬁned appropriately. Recall that for " > 0 the "-pseudospectrum is deﬁned as the open set

1 1 ".T/ C .T / > ; WD 2 Wk k " ° ± employing the convention that .T /1 for .T/ (see [35] for an k kD1 2 overview).

We allow the operators T , Tn, n N, to act in diﬀerent Hilbert spaces H, 2 Hn, n N, and require only convergence in so-called generalised strong resolvent 2 1 sense, i.e. the sequence of projected resolvents ..Tn / PHn /n2N shall converge 1 strongly to .T / PH in a common larger Hilbert space. The novelty of this paper lies in its general framework which is applicable to a wide range of operators T and approximating sequences .Tn/n2N.

1) We do not assumeselfadjointnessas in [29, Section VIII.7], [37, Section 9.3], or boundedness of the operators as in [34, 36, 17, 13] (see also [14] for an overview).

2) The operators may have non-empty essential spectrum, in contrast to the global spectral exactness results for operators with compact resolvents [3, 27, 38].

3) Theresults are applicable, butnot restricted to, the domain truncation method for diﬀerential operators [26, 9, 10, 11, 28, 15] and to the Galerkin (ﬁnite section) method [24, 8, 32, 25, 5, 7, 6].

4) Our assumptions are weaker than the convergence in operator norm [21] or in (generalised) norm resolvent sense [4, 20].

Regarding convergence of spectra (see (1.1)), the aim is to establish local spectral exactness of the approximation .Tn/n2N of T , i.e.

(1) local spectral inclusion: for every .T/ K there exists a sequence 2 \ .n/n2N of n .Tn/ K, n N, with n as n ; 2 \ 2 ! !1

(2) no spectral pollution: if there exists a sequence .n/n2I of n .Tn/ K, 2 \ n I , with n as n I , n , then .T/ K. 2 ! 2 !1 2 \ Concerning pseudospectra (see (1.2)), we deﬁne local "-pseudospectral exactness, -inclusion, -pollution in an analogous way by replacing all spectra by the closures of "-pseudospectra. Local convergence of spectra and pseudospectra 1053

In general, spectral exactness is a major challenge for non-selfadjoint problems. In the selfadjoint case, it is well known that generalised strong resolvent convergence implies spectral inclusion, and if the resolvents converge even in norm, then spectral exactness prevails [37, Section 9.3]. In the non-selfadjoint case, norm resolvent convergence excludes spectral pollution; however, the approximation need not be spectrally inclusive [23, Section IV.3]. Stability problems are simpler when passing from spectra to pseudospectra; in particular, they converge ("-pseudospectral exactness) under generalised norm resolvent convergence [4, Theorem 2.1]. However, if the resolvents converge only strongly, "-pseudospectral pollution may occur. In the two main results (Theorems 2.3, 3.6) we prove local spectral exactness outside the limiting essential spectrum ess..Tn/n2N/, and local "-pseudospectral exactness outside the limiting essential "-near spectrum ƒess;"..Tn/n2N/. The notion of limiting essential spectrum was introduced by Boulton, Boussaïd and Lewin in [8] for Galerkin approximations of selfadjoint operators. Here we gen- eralise it to our more general framework,

there exist I N and xn D.Tn/; n I; with ess..Tn/n2N/ C w 2 2 ; WD 2 W xn 1; xn 0; .Tn /xn 0 ´ k kD ! k k ! µ and further to pseudospectral theory,

there exist I N and xn D.Tn/; n I; with ƒess;"..Tn/n2N/ C w 2 2 : WD 2 W xn 1; xn 0; .Tn /xn " ´ k kD ! k k ! µ Outside these problematic parts, we prove convergence of the ("-pseudo-) spectra with respectto the Hausdorff metric. In the caseof pseudospectra, the problematic part is the whole complex plane if T has constant resolvent norm on an open set (see Theorem 3.8 and also [4]). The paper is organised as follows. In Section 2 we study convergence of spectra. First we prove local spectral exactness outside the limiting essential spectrum (Theorem 2.3). Then we establish properties of ess..Tn/n2N, including a spectral mapping theorem (Theorem 2.5) which implies a perturbation result for ess..Tn/n2N/ (Theorem 2.12). In Section 3 we address pseudospectra and prove local "-pseudospectral convergence (Theorem 3.6). Then we establish properties of the limiting essential "-near spectrum ƒess;"..Tn/n2N/ including a perturbation result (Theorem 3.15). In the final Section 4, applications to the Galerkin method of block-diagonally dominant matrices and to the domain truncation method of perturbed constant-coefficient PDEs are studied. 1054 S. Bögli

Throughout this paper we denote by H0 a separable infinite-dimensional Hilbert space. The notations and ; refer to the norm and scalar prod- kk h i uct of H0. Strong and weak convergence of elements in H0 is denoted by xn x w ! and xn 0, respectively. The space L.H/ denotes the space of all bounded op- ! erators acting in a Hilbert space H. Norm and strong operator convergence in s L.H/ is denoted by Bn B and Bn B, respectively. An identity operator ! ! is denoted by I ; scalar multiples I are written as . Let H;Hn H0, n N; 2 be closed subspaces and P PH H0 H, Pn PH H0 Hn; n N; D W ! D n W ! 2 be the orthogonal projections onto the respective subspaces and suppose that they s converge strongly, Pn P . Throughout, let T and Tn, n N, be closed, densely ! 2 defined linear operators acting in the spaces H, Hn, n N, respectively. The 2 domain, spectrum, point spectrum, approximate point spectrum and resolvent set of T are denoted by D.T/, .T/, p.T/, app.T/ and %.T /, respectively, and the Hilbert space adjoint operator of T is T . For non-selfadjoint operators there exist (at least) five different definitions for the essential spectrum which all coincide in the selfadjoint case; for a discussion see [16, Chapter IX]. Here we use

there exists .xn/n2N D.T/ with ess.T/ C w ; WD 2 W xn 1; xn 0; .T /xn 0 ´ k kD ! k k ! µ which corresponds to k 2 in [16]. The remaining spectrum dis.T/ D WD .T/ ess.T/ is called the discrete spectrum. For a subset C we denote n z z . Finally, for two compact subsets ;† C, their Haus- WD ¹NW 2 º dorﬀ distance is dH .;†/ max sup dist.z;†/; sup dist.z;/ where WD ¹ z2 z2† º dist.z;†/ infw2† z w . WD j j

2. Local convergence of spectra

In this section we address the problem of local spectral exactness. In Subsec- tion 2.1 we introduce the limiting essential spectrum ess..Tn/n2N and state the main result (Theorem 2.3). Then we establish properties of ess..Tn/n2N in Sub- section 2.2, including a spectral mapping theorem (Theorem 2.5) which implies a perturbation result for ess..Tn/n2N/ (Theorem 2.12). At the end of the section, in Subsection 2.3, we prove the main result and illustrate it for the example of Galerkin approximations of perturbed Toeplitz operators.

2.1. Main convergence result. The following deﬁnition of generalised strong and norm resolvent convergence is due to Weidmann [37, Section 9.3] who con- siders selfadjoint operators. Local convergence of spectra and pseudospectra 1055

Deﬁnition 2.1. i) The sequence .Tn/n2N is said to converge in generalised strong gsr resolvent sense to T , denoted by Tn T , if there exist n0 N and 0 ! 2 2 %.Tn/ %.T / with nn0 \

T 1 s 1 .Tn 0/ Pn .T 0/ P; n : ! !1

ii) The sequence .Tn/n2N is said to converge in generalised norm resolvent gnr sense to T , denotedby Tn T , if there exist n0 N and 0 %.Tn/ %.T / ! 2 2 nn0 \ with 1 1 T .Tn 0/ Pn .T 0/ P; n : ! !1 The following deﬁnition generalises a notion introduced in [8] for the Galerkin method of selfadjoint operators.

Deﬁnition 2.2. The limiting essential spectrum of .Tn/n2N is deﬁned as

there exist I N and xn D.Tn/; n I; with ess..Tn/n2N/ C w 2 2 : WD 2 W xn 1; xn 0; .Tn /xn 0 ´ k kD ! k k ! µ The following theorem is the main result of this section. We characterise regions where approximating sequences .Tn/n2N are locally spectrally exact and establish spectral convergence with respect to the Hausdorﬀ metric.

gsr gsr Theorem 2.3. i) Assume that Tn T and T T . Then spectral pollution is ! n ! conﬁned to the set ess..Tn/n2N/ ess..T /n2N/ ; (2.1) [ n and for every isolated .T/ that does notbelong to the setin (2.1), there exists 2 a sequence of n .Tn/; n N; with n ; n . 2 gsr 2 ! !1 ii) Assume that Tn T and Tn; n N; all have compact resolvents. Then ! 2 claim i) holds with (2.1) replaced by the possibly smaller set

ess..Tn /n2N/ : (2.2)

iii) Suppose that the assumptions of i) or ii) hold, and let K C be a compact subset such that K .T/ is discrete and belongs to the interior of K. If the \ intersection of K with the set in (2.1) or (2.2), respectively, is contained in .T/, then

dH ..Tn/ K;.T/ K/ 0; n : \ \ ! !1 1056 S. Bögli

2.2. Properties of the limiting essential spectrum. In this subsection we establish properties that the limiting essential spectrum shares with the essential spectrum (see [16, Sections IX.1,2]). The following result follows from a standard diagonal sequence argument; we omit the proof.

Proposition 2.4. The limiting essential spectrum ess ..Tn/n2N/ is a closed subset of C.

The limiting essential spectrum satisﬁes a mapping theorem.

Theorem 2.5. Let 0 N %.Tn/ and 0. Then the following are 2 n2 ¤ equivalent: T (1) ess..Tn/n2N/; 2 1 1 (2) . 0/ ess...Tn 0/ /n2N/. 2

Proof. (1) (2). Let ess ..Tn/n2N/. By Definition 2.2 of the limiting H) 2 essential spectrum, there exist I N and xn D.Tn/; n I; such that xn 1, w 2 2 k kD xn 0 and .Tn /xn 0. Note that .Tn 0/xn 0 0, hence ! k k! k k ! j j¤ there exists N N such that .Tn 0/xn > 0 for every n I with n N . 2 k k 2 Define .Tn 0/xn yn ; n I; n N: WD .Tn 0/xn 2 k k Then yn 1 and k kD .Tn /xn . 0/xn w yn 0; n : D .Tn 0/xn C .Tn 0/xn ! !1 k k k k Moreover, we calculate 1 1 1 xn . 0/ .Tn 0/xn .Tn 0/ . 0/ yn D .T /x n 0 n k k 1 .Tn /xn 0 k k 0; n : D j j .Tn 0/xn ! !1 k k 1 1 This implies . 0/ ess...Tn 0/ /n2N/. 2 (2) (1). It is easy to check that if there exist an infinite subset I N and H) w 1 1 yn Hn, n I , with yn 1, yn 0 and ..Tn 0/ . 0/ /yn 0, 2 2 k kD ! k k! then 1 .Tn 0/ yn xn 1 ; n I; WD .Tn 0/ yn 2 w k k satisfy xn 1, xn 0 and .Tn /xn 0. k kD ! k k! Local convergence of spectra and pseudospectra 1057

Remark 2.6. For the Galerkin method, Theorem 2.5 is diﬀerent from the spectral mapping theorem [8, Theorem 7] for semi-bounded selfadjoint operators. Whereas in Theorem 2.5 the resolvent of the approximation, for instance 1 .PnT R.Pn/ 0/ , is considered, the result in [8] is formulated in terms of the j 1 approximation of the resolvent, i.e. Pn.T 0/ R.P /, which is in general not j n easy to compute.

The essential spectrum is contained in its limiting counterpart.

gsr Proposition 2.7. i) Assume that Tn T . Then ess.T/ ess..Tn/n2N/: ! gnr ii) If Tn T , then ess.T/ ess ..Tn/n2N/. ! D For the proof we use the following elementary result.

gsr Lemma 2.8. Assume that Tn T . Then for all x D.T/ there exists a sequence ! 2 of elements xn D.Tn/; n N; with xn 1, n N, and 2 2 k kD 2

xn x Tnxn T x 0; n : (2.3) k kCk k ! !1 gsr Proof. By Deﬁnition 2.1 i) of Tn T , there exist n0 N and 0 %.T / such ! 2 2 that 0 %.Tn/, n n0, and 2 1 s 1 .Tn 0/ Pn .T 0/ P; n : (2.4) ! !1 Let x D.T/ and deﬁne 2 1 yn .Tn 0/ Pn.T 0/x D.Tn/; n n0: WD 2 s Then, using Pn P and (2.4), it is easy to verify that yn x 0 and ! k k ! Tnyn T x 0. In particular, there exists n1 n0 such that yn 0 for k k ! ¤ all n n1. Now (2.3) follows for arbitrary normalised xn D.Tn/, n < n1, and 2 xn yn= yn , n n1. WD k k

Proof of Proposition 2.7. i) Let ess.T/. By deﬁnition, there exist an inﬁnite 2 w subset I N and xk D.T/; k I; with xk 1, xk 0 and 2 2 k kD !

.T /xk 0; k : (2.5) k k ! !1 1058 S. Bögli

gsr Let k I be ﬁxed. Since Tn T , Lemma 2.8 implies that there exists a sequence 2 ! of elements xkIn D.Tn/; n N; such that xkIn 1, xkIn xk 0 2 2 k k D k k ! and TnxkIn T xk 0 as n . Let .nk/k2I be a sequence such that k k ! ! 1 nkC1 > nk; k I; and 2 1 1 xkIn xk < ; Tn xkIn T xk < ; k I: (2.6) k k k k k k k k k 2

w w Deﬁne xk xkIn D.Tn /; k I: Then (2.6) and xk 0 imply xk 0 Q WD k 2 k 2 ! Q ! as k I , k . Moreover, (2.5) and (2.6) yield .Tn /xk 0 as k I , 2 ! 1 k k Q k! 2 k . Altogether we have ess ..Tn/n2N/. !1 2 ii) ess.T/ ess ..Tn/n2N/ follows from i). Let ess ..Tn/n2N/. By the gnr 2 assumption Tn T , there exist n0 N and 0 %.T/ with 0 %.Tn/, ! 1 2 1 2 2 n n0, and .Tn 0/ Pn .T 0/ P . The mapping result established ! 1 1 in Theorem 2.5 implies . 0/ ess...Tn 0/ /nn0 /. So there are an 2 w inﬁnite subset I N and xn Hn, n I , with xn 1, xn 0 and 2 2 k kD ! 1 1 .Tn 0/ . 0/ xn 0; n I; n : ! 2 !1 Moreover, in the limit n we have !1 1 1 1 1 .T 0/ P . 0/ xn .Tn 0/ Pn . 0/ xn 1 1 .Tn 0/ Pn .T 0/ P 0: C !

Hence

1 1 1 0 . 0/ ess..T 0/ P/ ess..T 0/ / 0 : ¤ 2 [¹ º

Now ess.T/ follows from the mapping theorem [16, Theorem IX.2.3, k=2] 2 for the essential spectrum.

Now we study sequences of operators and perturbations that are compact or relatively compact in a sense that is appropriate for sequences. We use Stummel’s notion of discrete compactness of a sequence of bounded operators (see [33, Deﬁnition 3.1.(k)]).

Definition 2.9. Let Bn L.Hn/, n N. The sequence .Bn/n2N is said to be 2 2 discretely compact if for each infinite subset I N and each bounded sequence of elements xn Hn; n I; there exist x H and an infinite subset I I so 2 2 2 z that xn x 0 as n I , n . k k! 2 z !1 Local convergence of spectra and pseudospectra 1059

Proposition 2.10. i) If Tn L.Hn/, n N, are so that .Tn/n2N is a discretely 2 2 compact sequence and .Tn Pn/n2N is strongly convergent, then

ess..Tn/n2N/ 0 : D¹ º

If, in addition, .TnPn/n2N is strongly convergent, then

ess..T /n2N/ 0 : n D¹ º 1 ii) If there exists 0 n2N %.Tn/ such that ..Tn 0/ /n2N is a discretely 2 1 compact sequence and ..T 0/ Pn/n2N is strongly convergent, then Tn

ess..Tn/n2N/ : D ; 1 If, in addition, ..Tn 0/ Pn/n2N is strongly convergent, then ess..T /n2N/ : n D ; For the proof we need the following lemma. Claim ii) is the “discrete” analogue for operator sequences of the property of operators to be completely continuous.

s Lemma 2.11. Let Bn L.Hn/, n N, and B0 L.H0/ with B Pn B . 2 2 2 n ! 0 Consider an inﬁnite subset I N and elements x H0 and xn Hn, n I , w 2 2 2 such that xn x as n I , n . ! 2 !1 w i) We have x H and Bnxn B0x H as n I , n . 2 ! 2 2 !1

ii) If .Bn/n2N is discretely compact, then Bnxn B0x as n I , n . ! 2 !1

Proof. i) First note that, for any z H0, we have 2

xn;z xn;Pnz x;Pz Px;z ; n I; n ; h i D h i ! h i D h i 2 !1 w and hence xn P x. By the uniqueness of the weak limit, we obtain x ! w D P x H. The weak convergence Bnxn B0x is shown analogously, and also 2 w ! Bnxn PnBnxn PB0x which proves B0x PB0x H. D ! D 2 ii) Assume that there exist an inﬁnite subset I0 I and " > 0 such that

Bnxn B0x > "; n I0: (2.7) k k 2 1060 S. Bögli

Since the sequence .xn/n2I0 is bounded and .Bn/n2N is discretely compact, by Deﬁnition 2.9 there exists an inﬁnite subset I I0 such that .Bnxn/ H0 z n2Iz is convergent (in H0) to some y H. Then, by claim i), the strong convergence s 2 w w B Pn B and the weak convergence xn x imply Bnxn B0x H as n ! 0 ! ! 2 n I , n . By the uniqueness of the weak limit, we obtain y B0x, and 2 z ! 1 D therefore .Bnxn/n2Iz converges to B0x. The obtained contradiction to (2.7) proves the claim.

Proof of Proposition 2.10. i) Let I N be an inﬁnite subset, and let xn D.Tn/, w 2 n I , satisfy xn 1 and xn 0. Lemma 2.11 ii) implies Tnxn 0. Now the 2 k kD ! ! ﬁrst claim follows immediately.

Now assume that, in addition, .TnPn/n2N is strongly convergent. Then, by [3, Proposition 2.10], the sequence .Tn /n2N is discretely compact. Now the second claim follows analogously as the ﬁrst claim. ii) The assertion follows from i) and the mapping result in Theorem 2.5; note 1 that . 0/ 0 for all C. ¤ 2 The limiting essential spectrum is invariant under (relatively) discretely compact perturbations.

Theorem 2.12. i) Let Bn L.Hn/; n N. If the sequence .Bn/n2N is discretely 2 2 compact and .Bn Pn/n2N is strongly convergent, then

ess ..Tn Bn/n2N/ ess ..Tn/n2N/: C D

If, in addition, .BnPn/n2N is strongly convergent, then

ess...Tn Bn/ /n2N/ ess..T /n2N/ : C D n

ii) For n N let An be a closed, densely deﬁned operator in Hn. If there exists 2 0 N %.Tn/ N %.An/ such that the sequence 2 n2 \ n2 T T 1 1 ..Tn 0/ .An 0/ /n2N 1 1 is discretely compact and ...Tn 0/ .An 0/ / Pn/n2N is strongly convergent, then

ess..Tn/n2N/ ess..An/n2N/: D 1 1 If, in addition, ...Tn 0/ .An 0/ /Pn/n2N is strongly convergent, then ess..T /n2N/ ess..A /n2N/ : n D n Local convergence of spectra and pseudospectra 1061

Proof. i) Let I N be an infinite subset, and let xn D.Tn/, n I , satisfy w 2 2 xn 1 and xn 0. Lemma 2.11 ii) implies Bnxn 0. Now the first claim k k D ! ! follows immediately. Now assume that, in addition, .BnPn/n2N is strongly convergent. Then, by [3, Proposition 2.10], the sequence .Bn /n2N is discretely compact. Now the second claim follows analogously as the first claim. ii) By i), we have

1 1 ess...Tn 0/ /n2N/ ess...An 0/ /n2N/: D Now the ﬁrst claim follows from the mapping result in Theorem 2.5. 1 1 If, in addition, ...Tn / .An / /Pn/n2N is strongly convergent, then 1 1 [3, Proposition 2.10] implies that ...Tn 0/ .An 0/ / Pn/n2N is discretely compact. Now the second claim follows analogously.

2.3. Proof of local spectral convergence result and example. In this subsection we prove the local spectral exactness result in Theorem 2.3 and then illustrate it for the Galerkin method of perturbed Toeplitz operators. First we establish relations of the limiting essential spectrum with the following two notions of limiting approximate point spectrum and region of boundedness (introduced by Kato [23, Section VIII.1]).

Deﬁnition 2.13. The limiting approximate point spectrum of .Tn/n2N is deﬁnedas

there exist I N and xn D.Tn/; n I; with app..Tn/n2N/ C 2 2 ; WD 2 W xn 1; .Tn /xn 0 ´ k kD k k ! µ and the region of boundedness of .Tn/n2N is N C there exist n0 with %.Tn/; n n0; b..Tn/n2N/ 1 2 2 : WD 2 W . .Tn / /nn bounded ´ k k 0 µ The following lemma follows easily from Deﬁnitions 2.2 and 2.13.

Lemma 2.14. i) We have ess..Tn/n2N/ app..Tn/n2N/: ii) In general,

C b..Tn/n2N/ app..Tn/n2N/ app..T /n2N/ : n D [ n

If Tn; n N; all have compact resolvents, then 2 C b ..Tn/n2N/ app..Tn/n2N/ app..T /n2N/ : n D D n 1062 S. Bögli

Under generalised strong resolvent convergence we obtain the following relations. gsr Proposition 2.15. i) If Tn T , then ! app.T/ app..Tn/n2N/: gsr ii) If Tn T , then ! ess..T /n2N/ app..T /n2N/ ess..T /n2N/ p.T / : n n n [ gsr gsr iii) If Tn T and T T , then ! n ! app..Tn/n2N/ ess..Tn/n2N/ p.T /; D [ app..T /n2N/ ess..T /n2N/ p.T / : n D n [ Proof. i) The proof is analogous to the proof of Proposition 2.7; the only differ- ence is that here weak convergence of the considered elements is not required. ii) The first inclusion follows from Lemma 2.14 i). N Let app..Tn /n2N/ . Then there exist an infinite subset I and 2 xn D.T /, n I , with xn 1 and .T / 0 as n . Since .xn/n2I 2 n 2 k kD k n N k! !1 is a bounded sequence and H is weakly compact, there exists I I such that 0 z .xn/n2Iz converges weakly to some x H0. If x 0, then ess..Tn /n2N/ . 2gsr D 2 Now assume that x 0. Since Tn T , there exists 0 b ..Tn/n2N/ %.T/ 1 ¤ s !1 2 \ such that .Tn 0/ Pn .T 0/ P . The convergence .T /xn 0 ! k n N k ! implies

.T 0/xn . 0/xn yn; with yn .T /xn 0; n ; n D N C WD n N ! !1 hence

1 1 1 1 .T 0/ xn . 0/ xn yn; yn . 0/ .T 0/ yn; n D N Q Q WD N n 1 1 yn 0 .Tn 0/ yn 0; n : k Q k j j k kk k ! !1 w w Since x x, we obtain .T /1x . /1x as n I , n . On n n 0 n N 0 z ! ! 1 2w ! 11 the other hand, Lemma 2.11 i) yields x H and .Tn 0/ xn .T 0/ x. 2 1 ! 1 By the uniqueness of the weak limit, we obtain .T 0/ x .N 0/ x, 1 1 D hence . 0/ p..T 0/ /. This yields p.T / . N 2 2 iii) The second equality follows from claim ii), from p.T / app.T / and from claim i) (applied to T ;Tn ). Now we obtain the ﬁrst equality by replacing T ;Tn by T;Tn. Local convergence of spectra and pseudospectra 1063

The limiting essential spectrum is related to the region of boundedness as follows.

gsr gsr Proposition 2.16. i) If Tn T and T T , then ! n ! C b ..Tn/n2N/ p.T/ ess ..Tn/n2N/ p.T / ess..T /n2N/ ; n D [ [ [ n b..Tn/n2N/ %.T/ .C .ess..Tn/n2N/ ess..T /n2N/ // %.T /: \ D n [ n \ gsr ii) If Tn T and Tn; n N; all have compact resolvents, then ! 2 C b ..Tn/n2N/ p.T / ess..T /n2N/ ; n [ n b..Tn/n2N/ %.T / .C ess..T /n2N/ / %.T /: \ D n n \ Proof. i) The identities follow from Lemma 2.14 ii) and Proposition 2.15 iii). ii) The claims follow from the second part of Lemma 2.14 ii) and Proposi- tion 2.15 ii).

The local spectral convergence result (Theorem 2.3) relies on the following result from [3].

gsr Theorem 2.17 ([3, Theorem 2.3]). Suppose that Tn T . ! i) For each .T/ such that for some " > 0 we have 2

B"./ b ..Tn/n2N/ %.T /; n¹ º \

there exist n .Tn/; n N; with n as n . 2 2 ! !1 ii) No spectral pollution occurs in b ..Tn/n2N/.

Proof of Theorem 2.3. i) First note that the set in (2.1) is closed by Proposi- tion 2.4. If is an isolated point of .T/ and does not belong to the set in (2.1), then there exists " > 0 so small that

B"./ .C .ess..Tn/n2N/ ess..T /n2N/ // %.T /: n¹ º n [ n \

By Proposition 2.16 i), the right hand side coincides with b..Tn/n2N/ %.T /: \ Now the claims follow from Theorem 2.17. ii) The proof is analogous to i); we use claim ii) of Proposition 2.16. iii) Assume that the claim is false. Then there exist ˛ > 0, an inﬁnite subset I N and n K, n I , such that one of the following holds: 2 2 1064 S. Bögli

(1) n .Tn/ and dist.n;.T/ K/>˛ for every n I ; 2 \ 2 (2) n .T/ and dist.n;.Tn/ K/>˛ for every n I . 2 \ 2 Note that, in both cases (1) and (2), the compactness of K implies that there exist K and an inﬁnite subset J I such that .n/n2J converges to . 2 First we consider case (1). There are n .Tn/, n J , with n K. 2 2 ! 2 Since K does not contain spectral pollution by the assumptions, we conclude .T/ K. Hence 2 \

n dist.n;.T/ K/>˛; n J; j j \ 2 a contradiction to n . ! Now assume that (2) holds. The closedness of .T/ K yields .T/ K, \ 2 \ and the latter set is discrete by the assumptions. So there exists n0 N so that 2 n for all n J with n n0. In addition, by the above claim i) or ii), D 2 respectively, there exist n .Tn/, n N, so that n as n . Since 2 2 ! ! 1 .T/ K is in the interior of K by the assumptions, there exists n1 N so that 2 \ 2 n K for all n n1. So we conclude that, for all n J with n max n0; n1 , 2 2 ¹ º

n dist.;.Tn/ K// dist.n;.Tn/ K/> ˛; j j \ D \ a contradiction to n . This proves the claim. ! It is well known that truncating a Toeplitz operator (and compact perturbations of it) to finite sections is not a spectrally exact process but the pseudospectra converge in Hausdorff metric, see [5, 30]and[6, Theorem 3.17, Corollary 3.18 (b)] (where non-strict inequality in the definition of pseudospectra is used). In the following example we illustrate Theorem 2.3 for the Galerkin method of a compact perturbation of a Toeplitz operator using the perturbation result for the limiting essential spectrum (Theorem 2.12).

2 Example 2.18. Denote by ek k N the standard orthonormal basis of l .N/. ¹ W 2 º Let T L.l2.N// be the Toeplitz operator deﬁned by the so-called symbol 2 k f.z/ akz ; z C; WD Z 2 kX2 where ak C, k Z, are chosen so that f is continuous. This means that, with 2 2 1 respect to ek k N , the operator T has the matrix representation .Tij / ¹ W 2 º i;j D1 with

Tij Tej ;ei aij ; i;j N: WD h iD 2 Local convergence of spectra and pseudospectra 1065

The set [email protected]// is called symbol curve. Given [email protected]//, we deﬁne … the winding number I.f; / to be the winding number of [email protected]// about in the usual positive (counterclockwise) sense. The spectrum of T is, by [6, Theorem 1.17], given by

.T/ [email protected]// [email protected]// I.f; / 0 : D [¹ … W ¤ º 2 For n N, let Pn be the orthogonal projection of l .N/ onto Hn 2 s WD span ek k 1; : : : ; n . It is easy to see that Pn I . For a compact operator ¹ W2 D º ! S L.l .N//, let A T S and deﬁne An PnA H , n N. We claim that 2 WD C WD j n 2 the limiting essential spectra satisfy

ess..An/n2N/ ess..A /n2N/ .T/ .A/ (2.8) [ n I hence, by Theorem 2.3, no spectral pollution occurs for the approximation .An/n2N of A, and every isolated .A/ .T/ is the limit of a sequence 2 n .n/n2N with n .An/, n N. 2 2 s N To prove these statements, deﬁne Tn PnT Hn , n . Clearly, TnPn T , WD j 2 gsr gsr ! s s s AnPn A and Tn Pn T , AnPn A . Hence Tn T , An A and gsr! gsr ! ! ! ! T T , A A . By [6, Theorem 2.11], %.T / b ..Tn/n2N/. Using n ! n ! Proposition 2.16 i), we obtain

ess..Tn/n2N/ ess..T /n2N/ C b..Tn/n2N/: [ n n The perturbation result in Theorem 2.12 i) implies

ess..An/n2N/ ess..A /n2N/ ess..Tn/n2N/ ess..T /n2N/ : [ n D [ n So, altogether we arrive at the ﬁrst inclusion in (2.8). By [6, Theorem 1.17], ess.T/ ess.T / [email protected]// and for .T/ [email protected]// the operator [ D 2 n T is Fredholm with index ind.T / I.f; / 0. This means that D ¤ .T/ is equal to the set e4.T/ deﬁned in [16, Chapter IX], one of the (in general not equivalent) characterisations of essential spectrum. This set is invariant under compact perturbations by [16, Theorem IX.2.1], hence e4.T/ e4.A/ .A/, D which proves the second inclusion in (2.8). The rest of the claim follows from Theorem 2.3. For a concrete example, let

a3 7; a2 8; a1 1; a2 15; a3 5; D D D D D

ak 0; k Z 3; 2; 1; 2; 3 : D 2 n¹ º 1066 S. Bögli

(a) Symbol curve (red) corresponding to T and winding number in each component.

(b) Spectra and pseudospectra for n 10 (top), D n 50 (middle), n 100 (bottom). D D Figure 1. Spectra (eigenvalues as blue dots) and "-pseudospectra (" 2;1;21;:::;25) D of the truncated n n matrices An of the perturbed Toeplitz operator A T S. D C Local convergence of spectra and pseudospectra 1067

The symbol curve [email protected]// is shown in Figure 1 in red. The spectrum of the corresponding Toeplitz operator T consists of the symbol curve together with the connected components with winding numbers 1; 2; 1, see Figure 1 (a). So the resolvent set %.T / is the union of the connected components with winding number 0, which are denoted by 1 and 2 in Figure 1 (a). Now we add the compact operator S with matrix representation

1 20; i j 10; .Sij /i;j D1;Sij D WD ´0; otherwise:

By the above claim, the Galerkin approximation .An/n2N of A T S does WD C not produce spectral pollution, and every accumulation point of .An/, n N, in 2 1 2 belongs to .A/. Figure 1 (b) suggests that two such accumulation points [ exist in 1 and six in 2. Note that although the points in .T/ .A/ are not approximated by the Galerkin method, the resolvent norm diverges at these points, see Figure 1 (b). This is justified by [7, Proposition 4.2], which implies that for every .A/ 2 and every " > 0 there exists n;" N with ".An/, n n;" (see also 2 2 Theorem 3.3 below). Moreover, for any " > 0, in the limit n the closed "- ! 1 pseudospectrum ".An/ converges to ".A/ ".T/ with respect to the Hausdorff [ metric; this follows from [6, Corollary 3.18 (b)] and since every bounded Hilbert 1 space operator B satisfies ".B/ C .B / 1=" by e.g. [5, D ¹ 2 Wk k º Proposition 6.1].

3. Local convergence of pseudospectra

In this section we establish special properties and convergence of pseudospectra. Subsection 3.1 contains the main pseudospectral convergence result (Theo- rem 3.6). We also study the special case of operators having constant resolvent norm on an open set (Theorem 3.8). In Subsection 3.2 we provide properties of the limiting essential "-near spectrum ƒess;"..Tn/n2N/ including a perturbation result (Theorem 3.15), followed by Subsection 3.3 with the proofs of the results stated in Subsection 3.1.

3.1. Main convergence result. We ﬁx an " > 0.

Deﬁnition 3.1. Deﬁne the "-approximate point spectrum of T by

app;".T/ C there exists x D.T/ with x 1; .T /x <" : WD ¹ 2 W 2 k kD k k º 1068 S. Bögli

The following properties are well-known, see for instance [35, Chapter 4] and [12].

Lemma 3.2. i) The sets ".T/, app;".T/ are open subsets of C. ii) We have

".T/ app;".T/ .T/ app;".T/ app;".T / ; D [ D [ and app;".T/ .T/ app;".T / .T/: n D n If T has compact resolvent, then

".T/ app;".T/ app;".T / : D D iii) For ">"0 > 0,

"0 .T/ ".T/; ".T/ .T/: D ">0 \ iv) We have

C dist.;.T// < " ".T /; ¹ 2 W º with equality if T is selfadjoint.

In contrast to the spectrum, the "-pseudospectrum is always approximated under generalised strong resolvent convergence. For bounded operators and strong convergence, this was proved by Böttcher-Wolf in [7, Proposition 4.2] (where non- strict inequality in the deﬁnition of pseudospectrais used); claim i) is not explicitly stated but can be read oﬀ from the proof. Note that if T has compact resolvent, then claim i) holds for all ".T/ app;".T/ by Lemma 3.2 ii). 2 D gsr Theorem 3.3. Suppose that Tn T . ! i) For every app;".T/ and x D.T/ with x 1, .T /x <" there 2 2 k kD k k exist n N and xn D.Tn/, n n, with 2 2

app;".Tn/; xn 1; .Tn /xn < "; n n; 2 k kD k k

and xn x 0 as n . k k! !1 gsr ii) Suppose that, in addition, T T . Then for every ".T/ there exists n ! 2 n N such that ".Tn/, n n. 2 2 Local convergence of spectra and pseudospectra 1069

The following example illustrates that we cannot omit the additional assump- gsr tion Tn T in Theorem 3.3 ii). In particular, this is a counterexample for [9, ! gsr Theorem 4.4] where only Tn T is assumed. ! Example 3.4. Let T be the ﬁrst derivative in L2.0; / with Dirichlet boundary 1 condition,

Tf f 0; D.T/ f W 1;2.0; / f .0/ 0 : WD WD ¹ 2 1 W D º 2 We approximate T by a sequence of operators Tn in L .0; n/, n N, deﬁned by 2 0 1;2 Tnf f ; D.Tn/ f W .0; n/ f .0/ f.n/ : WD WD ¹ 2 W D º Note that C Re 0 .T/ ".T /: The operators i Tn, n N, are ¹ 2 W º D 2 selfadjoint. Hence

C Re < 0 ..Tn/n2N/ %.T /: ¹ 2 W º b \ Using that f D.T/ supp f compact is a core of T , [3, Theorem 3.1] implies gsr gsr ¹ 2 W º that Tn T . However, since i T is not selfadjoint, we obtain Tn Tn ! D ! T T . The selfadjointness of i Tn and Lemma 3.2 iv) imply ¤ ".Tn/ C dist.;.Tn//<" C Re <" ; n N: D¹ 2 W º¹ 2 W j j º 2 Therefore, for every C with Re >", we conclude ".T/ but 2 2 dist.;".Tn// Re " > 0; n N: 2 In order to characterise "-pseudospectral pollution, we introduce the following sets.

Deﬁnition 3.5. Deﬁne the essential "-near spectrum of T by w xn 1; xn 0; ƒess;".T/ C there exists xn D.T/; n N; with k kD ! ; WD 2 W 2 2 .T /xn " ´ k k ! µ and the limiting essential "-near spectrum of .Tn/n2N by

there exist I N and xn D.Tn/; n I; with ƒess;"..Tn/n2N/ C w 2 2 : WD 2 W xn 1; xn 0; .Tn /xn " ´ k kD ! k k ! µ The following theorem is the main result of this section. We establish local "-pseudospectral exactness and prove "-pseudospectral convergence with respect to the Hausdorﬀ metric in compact subsets of the complex plane where we have "-pseudospectral exactness. 1070 S. Bögli

gsr gsr Theorem 3.6. Suppose that Tn T and T T . ! n ! i) The sequence .Tn/n2N isan "-pseudospectrally inclusive approximation of T . ii) Deﬁne

ƒess;.0;" .ƒess;ı ..Tn/n2N/ ƒess;ı ..T /n2N/ /: WD \ n ı2[.0;" Then "-pseudospectral pollution is conﬁned to

ess..Tn/n2N/ ess..T /n2N/ ƒess;.0;" (3.1) [ n [ I

if the operators Tn, n N, all have compact resolvents, then it is restricted 2 to .ess..Tn/n2N/ ess..T /n2N/ / ƒess;.0;": (3.2) \ n [ iii) Let K C be a compact subset with

".T/ K ".T/ K : \ D \ ¤ ; If the intersection of K with the set in (3.1) or (3.2), respectively, is contained in ".T/, then

dH .".Tn/ K; ".T/ K/ 0; n : \ \ ! !1 Remark 3.7. If we compare Theorem 3.6 iii) with [4, Theorem 2.1] for generalised norm resolvent convergence, note that here we do no explicitly exclude the possi- bility that .T /1 is constant on an open subset U %.T /. How- 7! k k ;¤ ever, if the resolvent norm is equal to 1=" on an open set U , then U ".T/ \ D ; and hence the following Theorem 3.8 ii) implies that a compact set K with K ƒess;"..Tn/n2N/ ".T/ satisﬁes K U . So we implicitly exclude \ \ D ; the problematic region U .

In the following result we study operators that have constant resolvent norm on an open set. For the existence of such operators see [31, 4].

Theorem 3.8. Assume that there exists an open subset U %.T / such that ;¤ 1 .T /1 ; U: k kD " 2 i) We have

%.T / C .T K/ ƒess;".T/ ƒess;".T / : n C \ K compact\ Local convergence of spectra and pseudospectra 1071

gsr gsr ii) If Tn T and T T , then ! n ! %.T/ C .T K/ ƒess;"..Tn/n2N/ ƒess;"..T /n2N/ : n C \ n K compact\

Remark 3.9. Note that, by [16, Section IX.1, Theorems IX.1.3, 1.4],

ess.T/ ess.T / e3.T/ e4.T/ .T K/: [ D D C K compact\ 3.2. Properties of the limiting essential "-near spectrum

Proposition 3.10. i) The sets ƒess;".T/, ƒess;"..Tn/n2N/ are closed subsets of C. ii) We have

z ess.T /; z " ƒess;".T /; (3.3a) ¹ C W 2 j jD º

z ess..Tn/n2N/; z " ƒess;"..Tn/n2N/: (3.3b) ¹ C W 2 j jD º Proof. A diagonal sequence argument implies claim i), and claim ii) is easy to see.

Remark 3.11. The inclusions in claim ii) may be strict. In fact, for Shargorodsky’s example [31, Theorem 3.2] of an operator T with constant (1=" 1) resolvent D norm on anopenset, the compressions Tn ontothespanofthe ﬁrst 2n basis vectors satisfy

ess..Tn/n2N/ ess.T/ ; .T K/ : D D ; C D ; K compact\ Hence the left hand side of (3.3) is empty whereas the right hand side equals C by Theorem 3.8.

Analogously as ess.T/ ess..Tn/n2N/ (see Proposition 2.7), also the essential "-near spectrum is contained in its limiting counterpart.

gsr Proposition 3.12. i) Assume that Tn T . Then ƒess;".T/ ƒess;"..Tn/n2N/: ! gnr ii) If H0 H and Tn T , then ƒess;".T/ %.T / ƒess;"..Tn/n2N/ %.T /: D ! \ D \ 1072 S. Bögli

For the proof we use the following simple result.

gsr Lemma 3.13. Assume that Tn T . Suppose that there exist an infinite subset ! w I N and xn D.Tn/, n I , with xn 1 and xn 0. If . Tnxn /n2I is 2 w 2 k k D ! k k bounded, then Tnxn 0. ! 1 s Proof. Define yn Tnxn. Let 0 b..Tn/n2N %.T / satisfy .Tn 0/ Pn 1 WD 2 \ ! .T 0/ P . Since H0 is weakly compact, there exist y H0 and an infinite 2 subset I I such that .yn/ z converges weakly to y. We prove that y 0. z n2I D 1 w 1 Lemma 2.11 i) implies y H and .Tn 0/ yn .T 0/ y. Hence 1 2 w 1 ! xn .Tn 0/ yn 0xn .T 0/ y. The uniqueness of the weak limit D 1 ! yields .T 0/ y 0 and thus y 0. D D Proof of Proposition 3.12. i) The proof is analogous to the proof of i) of Proposi- tion 2.7.

ii) Using claim i), it remains to prove ƒess;"..Tn/n2N/ %.T / ƒess;".T/. \ Let ƒess;"..Tn/n2N/ %.T /. By Definition 3.5, there exist an infinite subset 2 \ w I N and xn D.Tn/, n I , with xn 1, xn 0 and .Tn /xn ". 2 2 k k D ! k k ! Since %.T /, [3, Proposition 2.16 ii)] implies that there exists n N such that 2 1 1 2 %.Tn/, n n, and .Tn / Pn .T / . Define I2 I n n 2 ! WD ¹ 2 W º and .Tn /xn wn Hn H; n I2: WD .Tn /xn 2 2 k k w Then wn 1 and wn 0 by Lemma 3.13. In addition, k kD !

1 1 1 .Tn / wn ; n I2; n : k kD .Tn /xn ! " 2 !1 k k Since, in the limit n I2, n ; 2 !1 1 1 1 1 .Tn / wn .T / wn .Tn / Pn .T / 0; jk kk kjk k ! 1 we conclude .T / wn 1=". Now deﬁne k k! 1 .T / wn D vn 1 .T/; n I2: WD .T / wn 2 2 k k w 1 1 Then vn 1 and vn 0. Moreover, .T /vn .T / wn ", k k D ! k kDk k ! hence ƒess;".T/. 2 Local convergence of spectra and pseudospectra 1073

Similarly as for ess..Tn/n2N/ (see Proposition 2.10), the set ƒess;"..Tn/n2N/ is particularly simple if the operators Tn, n N, or their resolvents form a discretely 2 compact sequence.

Proposition 3.14. i) If Tn L.Hn/, n N, are so that .Tn/n2N is a discretely 2 2 compact sequence and .Tn Pn/n2N is strongly convergent, then

ƒess;"..Tn/n2N/ C " : D¹ 2 W j jD º

If, in addition, .TnPn/n2N is strongly convergent, then

ƒess;"..T /n2N/ C " : n D¹ 2 W j jD º 1 ii) If there exists 0 n2N %.Tn/ %.T / such that ..Tn 0/ /n2N is a 2 \ 1 s 1 discretely compact sequence and .T 0/ Pn .T 0/ P , then T n !

ƒess;"..Tn/n2N/ : D ; 1 s 1 If, in addition, .Tn 0/ Pn .T 0/ P , then ! ƒess;"..T /n2N/ : n D ; Proof. i) The proof is similar to the one of Proposition 2.10 i).

ii) Assume that there exists ƒess;"..Tn/n2N/. Then there are an inﬁnite 2 subset I N and xn D.Tn/, n I , with 2 2 w xn 1; xn 0; .Tn /xn "; n I; n : k kD ! k k ! 2 !1 Deﬁne

yn .Tn /xn; n I: WD 2 w By Lemma 3.13, yn 0 as n I , n . Hence ! 2 !1 w .Tn 0/xn yn . 0/xn 0 D C ! and thus, by the assumptions and Lemma 2.11 ii),

1 xn .Tn 0/ .yn . 0/xn/ 0; n I; n : D C ! 2 !1

The obtained contradiction to xn 1, n I , proves the ﬁrst claim. k kD 2 1 The second claim is obtained analogously, using that ..T 0/ /n2N is n discretely compact by [3, Proposition 2.10]. 1074 S. Bögli

We prove a perturbation result for ƒess;"..Tn/n2N/ and, in claim ii), also for ess..Tn/n2N/; for the latter we use that the assumptions used here imply the assumptions of Theorem 2.12.

Theorem 3.15. i) Let Bn L.Hn/; n N. If the sequence .Bn/n2N is discretely 2 2 compact and .Bn Pn/n2N is strongly convergent, then

ƒess;" ..Tn Bn/n2N/ ƒess;" ..Tn/n2N/: C D

If, in addition, .BnPn/n2N is strongly convergent, then

ƒess;" ..Tn Bn/ /n2N ƒess;" .T /n2N : C D n

ii) Let S and Sn, n N, be linear operators in H and Hn, n N, with 2 2 D.T/ D.S/ and D.Tn/ D.Sn/, n N, respectively. Assume that there exist 2 0 N %.Tn/ %.T / and < 1 such that 2 n2 \ 0 1 1 (a) TS.T 0/ < 1 and Sn.Tn 0/ for all n N; k k k k 0 2 1 (b) the sequence .Sn.Tn 0/ /n2N is discretely compact; (c) for n , !1 1 s 1 .T 0/ Pn .T 0/ P; n ! 1 s 1 .Sn.Tn 0/ / Pn .S.T 0/ / P: !

Then the sums A T S and An Tn Sn, n N, satisfy WD C WD C 2 1 s 1 .A 0/ Pn .A 0/ P n ! and

ƒess;"..An/n2N/ ƒess;"..Tn/n2N/; ess..An/n2N/ ess..Tn/n2N/: (3.4) D D iii) Let S be a linear operator in H with D.T/ D.S/. If there exists 1 1 0 %.T / such that S.T 0/ < 1 and S.T 0/ is compact, then 2 k k

ƒess;".T S/ ƒess;".T /: C D Local convergence of spectra and pseudospectra 1075

Proof. i) The proof is analogous to the proof of Theorem 2.12 i). ii) The proof relies on the following claim, which we prove at the end.

Claim. We have 0 N %.An/ %.A/, the sequences 2 n2 \ 1 1 1 ..Tn 0/T .An 0/ /n2N; .Sn.An 0/ /n2N (3.5) are discretely compact and, in the limit n , !1 1 s 1 .A 0/ Pn .A 0/ P; (3.6a) n ! 1 1 s 1 1 ..Tn 0/ .An 0/ / Pn ..T 0/ .A 0/ / P; (3.6b) ! 1 s 1 .Sn.An 0/ / Pn .S.A 0/ / P: (3.6c) ! The second equality in (3.4) follows from the above Claim and Theorem 2.12 ii). Now let ƒess;"..An/n2N/. Then there exist an inﬁnite subset I N and 2 xn D.An/, n I , with 2 2 w xn 1; xn 0; .An /xn "; n I; n : k kD ! k k ! 2 !1 Deﬁne yn .An /xn; n I: WD w 2 By Lemma 3.13, we conclude yn 0 as n I , n . Then .An 0/xn w ! 2 ! 1 D yn . 0/xn 0 and thus, by the above Claim and Lemma 2.11 ii), C ! 1 Snxn Sn.An 0/ .yn . 0/xn/ 0; n I; n : D C ! 2 !1 Therefore, .Tn /xn .An /xn Snxn " and hence k k k kCk k ! 2 ƒess;"..Tn/n2N/. The reverse inclusion ƒess;"..Tn/n2N/ ƒess;"..An/n2N/ is proved analo- 1 gously, using that .Sn.Tn 0/ /n2N is discretely compact and 1 s 1 .Sn.Tn 0/ / Pn .S.T 0/ / P ! by assumptions (b), (c).

Proof of Claim. A Neumann series argument implies that, for every n N, we 2 have 0 %.An/ and 2 1 1 1 1 .An 0/ .Tn 0/ .I Sn.Tn 0/ / ; (3.7a) D C 1 1 1 1 .A 0/ .I .Sn.Tn 0/ / / .T 0/ ; (3.7b) n D C n 1 1 1 1 .Sn.An 0/ / .I .Sn.Tn 0/ / / .Sn.Tn 0/ / (3.7c) D C I 1076 S. Bögli for A, S, T we obtain analogous equalities. Now we apply [3, Lemma 3.2] 1 1 to B .S.T 0/ / and Bn .Sn.Tn 0/ / , n N; note that D D 2 1 b..Bn/n2N/ %.B/ by assumption (a). Hence we obtain 2 \ 1 1 s 1 1 .I .Sn.Tn 0/ / / Pn .I .S.T 0/ / / P: C ! C Now the strong convergences (3.6) follow from (3.7) and assumption (c). To prove discrete compactness of the sequences in (3.5), we use that

1 1 1 1 .Tn 0/ .An 0/ .An 0/ Sn.Tn 0/ ; D 1 1 1 1 Sn.An 0/ Sn.Tn 0/ .I Sn.Tn 0/ / : D C Now the claims are obtained by [3, Lemma 2.8 i), ii)] and using assump- 1 s 1 tions (a), (b), and .A 0/ Pn .A 0/ P by (3.6). n ! 4

iii) The assertion follows from claim ii) applied to Tn T , Sn S, n N. D D 2 3.3. Proofs of pseudospectral convergence results. First we prove the "- pseudospectral inclusion result.

gsr Proof of Theorem 3.3. i) The assumption Tn T and Lemma 2.8 imply that ! there are xn D.Tn/, n N, with xn 1, xn x 0, Tnxn T x 0. 2 2 k kD k k! k k! Hence there exists n N such that, for all n n, 2

.Tn /xn .T /x Tnxn T x xn x < ": k kk kCk k C j jk k

Therefore, app;".Tn/ ".Tn/ for all n n. 2 ii) By Lemma 3.2 ii), ".T/ app;".T/ app;".T / . Now the assertion D gsr[ gsr follows from claim i) and the assumptions Tn T , T T . ! n ! Now we conﬁne the set of pseudospectral pollution.

gsr gsr Proposition 3.16. Suppose that Tn T and T T . Let %.T / ! n ! 2 \ b..Tn/n2N/ and let n C, n N, satisfy n , n . Then 2 2 ! !1 1 1 1 M lim sup .Tn n/ lim sup .Tn / .T / WD n!1 k kD n!1 k kk kI if the inequality is strict, then

ƒ 1 ..Tn/n2N/ ƒ 1 ..Tn /n2N/ : 2 ess; M \ ess; M Local convergence of spectra and pseudospectra 1077

Proof. First we prove that

1 1 M lim sup .Tn n/ lim sup .Tn / : (3.8) D n!1 k kD n!1 k k 1 Since b..Tn/n2N/, we have C sup N .Tn / < . A Neumann 2 WD n2 k k 1 series argument yields that, for all n N so large that n < 1=C , 2 j j 1 1 1 1 C .Tn n/ .Tn / .I .n /.Tn / / : k kDk k 1 n C j j The ﬁrst resolvent identity implies

1 1 1 1 .Tn / .Tn n/ n .Tn n/ .Tn / jk kk kjj jk kk k C 2 n : j j1 n C j j The right hand side converges to 0 since n . This proves (3.8). ! The inequality

1 1 lim sup .Tn / .T / n!1 k kk k follows from Theorem 3.3 ii). 1 1 1 Now assume that M > .T / . First note that .T / .Tn / is k k n N selfadjoint and 1 2 1 1 .Tn / sup .Tn /N .Tn / y; y k k D kykD1h i 1 1 max app..T / .Tn / /: D n N Therefore, 1 1 M app...T / .Tn / /n2N/: 2 n N gsr gsr By the assumptions Tn T , T T and b..Tn/n2N/ %.T /, we obtain, ! n ! 2 \ using [3, Proposition 2.16],

1 1 s 1 1 .T / .Tn / Pn .T / .T / P; n : n N ! N !1 gsr 1 1 1 1 Moreover, [3, Lemma 3.2] yields .T / .Tn / .T / .T / . n N ! N Now Proposition 2.15 ii) implies that

2 1 1 1 1 M p..T / .T / / ess...T / .Tn / /n2N/: 2 N [ n N 2 1 1 First case. If M p..T / .T / /, then there exists y H with 2 N 2 y 1 such that k kD 0 ..T /1.T /1 M 2/y;y .T /1y 2 M 2: D h N iDk k So we arrive at the contradiction .T /1 M . k k 1078 S. Bögli

2 1 1 Second case. If M ess...Tn /N .Tn / /n2N/, then the mapping result 2 2 in Theorem 2.5 implies that 1=M ess...Tn /.Tn //N n2N/: Hence there N 2 D D exist an inﬁnite subset I and xn ..Tn /.Tn //N .Tn /, n I , w 2 2 2 with xn 1, xn 0 and ..Tn /.T / 1=M /xn 0. So we arrive k kD ! k n N k! at

2 1 .T /xn .Tn /.T /xn; xn ; n I; n : k n N k D h n N i ! M 2 2 !1 This implies ƒess;1=M ..Tn /n2N/ . 2 1 1 1 1 Since .T / .T / and .Tn / .T / , we k kDk N k k kDk n N k obtain analogously that ƒess;1=M ..Tn/n2N/. 2 Next we prove the "-pseudospectral exactness result.

Proof of Theorem 3.6. i) Let ".T/. Assume that the claim is false, i.e. 2

˛ lim sup dist.; ".Tn// > 0: (3.9) WD n!1

Choose ".T/ with < ˛=2. By Theorem 3.3 ii), there exists nQ N Q 2 j Q j 2 such that ".Tn/ ".Tn/, n nQ , which is a contradiction to (3.9). Q 2 ii) Choose C ".T/ outside the set in (3.1) or (3.2), respectively. Assume 2 n that it is a point of "-pseudospectral pollution, i.e. there exist an inﬁnite subset I N and n ".Tn/, n I , with n . By the choice of and 2 2 ! Proposition 2.16 i), we arrive at %.T / b..Tn/n2N/. 2 \ Since n ".Tn/, n I , we have 2 2

1 1 M lim sup .Tn n/ : WD n2I k k " n!1

1 Now, by Proposition 3.16 and using ƒess;.0;", we conclude that .T / … k kD 1=". ByTheorem 3.8 ii), the level set %.T / .T /1 1=" does not have ¹ 2 Wk kD º an open subset. Hence we arrive at the contradiction ".T/, which proves the 2 claim. iii) The proof is similar to the one of Theorem 2.3 iii). Assume that the claim is false. Then there exist ˛ > 0, an inﬁnite subset I N and n K, n I , such 2 2 that one of the following holds:

(1) n ".Tn/ and dist.n; ".T/ K/>˛ for every n I ; 2 \ 2 (2) n ".T/ and dist.n; ".Tn/ K/>˛ for every n I . 2 \ 2 Local convergence of spectra and pseudospectra 1079

Note that, in both cases (1) and (2), the compactness of K implies that there exist K and an inﬁnite subset J I such that .n/n2J converges to . 2 First we consider case (1). Claim ii) and the assumptions on K imply that ".T/ K and hence n dist.n; ".T/ K/ > ˛, n J , a 2 \ j j \ 2 contradiction to n . ! Now assume that (2) holds. The assumption ".T/ K ".T/ K implies \ D \ that .n/n2J ".T/ K and thus ".T/ K. Choose ".T/ K with \ 2 \ Q 2 \ < ˛=2. By Theorem 3.3 ii), there exists nQ N such that ".Tn/ K j Q j 2 Q 2 \ for every n nQ . Therefore n dist.n; ".Tn/ K/>˛ for every n J j Q j \ 2 with n nQ . Since n < ˛=2, we arrive at a contradiction. This j Q j ! j Q j proves the claim.

Finally we prove the result about operators that have constant resolvent norm on an open set.

Proof of Theorem 3.8. i) Let 0 U . By proceeding as in the proof of Theo- 2 rem 3.6 (with Tn T , n 0, n N, and M 1="), we obtain D D D 2 D

1 1 1 1 1 p..T 0/ .T 0/ / ess..T 0/ .T 0/ /; "2 2 [

2 1 1 and the secondcase 1=" ess..T 0/ .T 0/ / implies 0 ƒess;".T / . 2 2 1 1 2 If however 1=" p..T 0/ .T 0/ /, then there exists y H with 2 2 y 1 such that .T /1y 1=". Hence x .T /1y 0 satisfies k k D k k D WD ¤ .T /x = x ". Note that .T /x is a non-constant subharmonic k k k kD 7! k k function on C and thus satisfies the maximum principle. Therefore, in every open neighbourhood of 0 there exist points such that .T /x = x <" and thus k k k k 0 ".T/, a contradiction. 2 Since .T /1 .T /1 1=" for every U , we analogously k kDk N k D 2 obtain 0 ƒess;".T/. Sothere exists a sequence .xn/n2N D.T/ with xn 1, w 2 k kD xn 0 and .T 0/xn ". Define ! k k! .T 0/xn en ; n N: WD .T 0/xn 2 k k w Then en 1 and en 0 by Lemma 3.13 applied to Tn T . In addition, k k D1 ! 1 D .T 0/ en .T 0/ 1=". Analogously as in the proof of [4, k k!k k D Theorem 3.2], using the old results [19, Lemmas 1.1,3.0] by Globevnik and Vidav, one may show that 2 .T 0/ en 0; n (3.10) ! ! 1I 1080 S. Bögli note that [4, Theorem 3.2] was proved for complex uniformly convex Banach spaces and is thus, in particular, valid for Hilbert spaces (see [18] for a discussion about complex uniform convexity). Let C .T K/: Then there exists a compact operator 2 n K compact C K L.H/ such that %.T K/. The second resolvent identity applied twice 2 T 2 C yields

1 1 .T K / .T 0/ C 1 1 .T K / . K 0/.T 0/ D C C 1 1 1 .I .T K / . K 0//.T 0/ K.T 0/ D C C C 1 2 . 0/.I .T K / . K 0//.T 0/ : C C C C 1 1 1 Since K .I .T K / . K 0//.T 0/ K.T 0/ is compact z WD C C C w and hence completely continuous, the weak convergence e 0 yields Ke 0. n z n 1 ! 1 ! Using (3.10) in addition, we conclude ..T K / .T 0/ /en 0 and C ! hence 1 1 1 lim .T K / en lim .T 0/ en : n!1 k C kD n!1 k kD " Now deﬁne .T K /1e n N wn C 1 ; n : WD .T K / en 2 k C k w 1 1 Then wn 1, wn 0 and .T K /wn .T K / en ". k kD ! k C kDk C k ! Therefore, ƒess;".T K/. Theorem 3.15 i) applied to Tn T and Bn K 2 C D D yields ƒess;".T K/ ƒess;".T/. So arrive at ƒess;".T/. C D 2 In an analogous way as for (3.10), one may show that there exists a normalised w 2 sequence .fn/n2N H with fn 0 and .T 0/ fn 0. So, by proceeding ! ! as above, we obtain ƒess;".T / . 2 ii) The claim follows from claim i) and Proposition 3.12 i).

4. Applications and Examples

In this section we discuss applications to the Galerkin method for inﬁnite matrices (Subsection 4.1) and to the domain truncation method for diﬀerential operators (Subsection 4.2).

4.1. Galerkin approximation of block-diagonally dominant matrices. In this subsection we consider an operator A in l2.K/ (where K N or K Z) whose D D matrix representation (identiﬁed with A) with respect to the standard orthonormal Local convergence of spectra and pseudospectra 1081 basis ej j K can be split as A T S. Here T is block-diagonal, i.e. there ¹ W 2 º D C exist mk N with 2

mkmk T diag.Tk k K/; Tk C : D W 2 2 We further assume that D.T/ D.S/, D.T / D.S / and that there exists 1 0 %.T / such that S.T 0/ is compact and 2 1 1 S.T 0/ < 1; S .T 0/ < 1: (4.1) k k k k Deﬁne, for n N, 2 0 n mk; K Z; jn 8 D Jn mk: WD kDn WD ˆ X kD1 <1; K N; X D ˆ 2 Let Pn be the orthogonal: projection of l .K/ onto Hn span ej jn j Jn . s WD ¹ W º It is easy to see that Pn I . !

Theorem 4.1. Deﬁne An PnA H , n N. WD j n 2 gsr gsr i) We have An A and A A . ! n ! ii) The limiting essential spectra satisfy

ess..An/n2N/ ess..A /n2N/ [ n 1 C there exists I N with .Tn / ; n I; n D¹ 2 W k k !1 2 ! 1º ess.A/ D I

hence no spectral pollution occurs for the approximation .An/n2N of A, and for every isolated dis.A/ there exists a sequence of n .An/, n N, 2 2 2 with n as n . ! !1 iii) The limiting essential "-near spectrum satisﬁes

ƒess;"..An/n2N/

1 1 C there exists I N with .Tn / ; n I; n D 2 W k k ! " 2 !1 ° ± ƒess;".A/ D I hence if A does not have constant resolvent norm ( 1=") on an open set, D then .An/n2N is an "-pseudospectrally exact approximation of A. 1082 S. Bögli

Proof. First note that the adjoint operators satisfy A T S , and since, n D n C n by (4.1), S is T -bounded and S is T -bounded with relative bounds < 1, [22, Corollary 1] implies A T S . In addition, for any n N, D C 2 1 1 .Tn 0/ Pn Pn.T 0/ ; (4.2a) D 1 1 .T 0/ Pn Pn.T 0/ ; (4.2b) n D 1 1 Sn.Tn 0/ Pn PnS.T 0/ ; (4.2c) D 1 1 S .T 0/ Pn PnS .T 0/ ; (4.2d) n n D 1 1 .Sn.Tn 0/ / Pn D.S/ Pn.T 0/ S j D (4.2e) 1 Pn.S.T 0/ / D.S/: D j Now, using (4.2) everywhere, we check that the assumptions of Theorem 3.15 ii), and iii) are satisﬁed. (a) We readily conclude

1 1 Sn.Tn 0/ S.T 0/ < 1: (4.3) k kk k 1 1 N (b) The sequence of operators Sn.Tn 0/ PnS.T 0/ Hn , n , is 1 D s j 2 discretely compact since S.T 0/ is compact and Pn I . ! 1 s 1 (c) The strong convergence .Tn 0/ Pn .T 0/ follows immediately s ! from Pn I . In addition, since D.S / is a dense subset, using (4.3) we ! 1 s 1 obtain .Sn.Tn 0/ / Pn .S.T 0/ / . ! gsr Now Theorem 3.15 ii), iii) implies A A and n ! ess..An/n2N/ ess..Tn/n2N/; D

ƒess;"..An/n2N/ ƒess;"..Tn/n2N/; D

ƒess;".A/ ƒess;".T /: D 1 In addition, since S.T 0/ is assumed to be compact, [16, Theorem IX.2.1] yields ess.A/ ess.T/. D gsr In claim i) it is left to be shown that An A. To this end, we use that (4.2) s ! and Pn I imply ! 1 s 1 1 s 1 .Tn 0/ Pn .T 0/ ;Sn.Tn 0/ Pn S.T 0/ : ! ! Now the claim follows from (4.3) and the perturbation result [3, Theorem 3.3]. Local convergence of spectra and pseudospectra 1083

Note that Theorem 2.12 ii) implies ess..A /n2N/ ess..T /n2N/ . The n D n identities in claim ii) are obtained from

ess..Tn/n2N/ ess..T /n2N/ [ n N C there exists I with 1 D 2 W .Tn / ; n I; n ´ k k !1 2 !1 µ ess.T/: D Now the local spectral exactness follows from Theorem 2.3. The assertion in iii) follows from an analogous reasoning, using Theorem 3.6; note that if T does not have constant ( 1=") resolvent norm on an open set, then D ƒess;".T/ @".T/ ".T/ andhenceno "-pseudospectral pollution occurs.

Example 4.2. For points b; d C and sequences .aj /j 2N; .bj /j 2N, .cj /j 2N, 2 .dj /j 2N C with

aj ; bj b; cj 0; dj d; j ; j j !1 ! ! ! !1 deﬁne an unbounded operator A in l2.N/ by

a1 b1

0c1 d1 b2 1 A B c2 a2 b3 C ; WD B C B :: C B c3 d2 :C B C B :: :: C B : :C B C @ A 2 2 D.A/ .xj /j 2N l .N/ aj x2j 1 < : WD 2 W j j 1 j 2N ° X ±

2 For n N let Pn be the orthogonal projection of l .N/ onto the ﬁrst 2n basis 2 vectors, and deﬁne An PnA R.P /. Using Theorem 4.1, we show that WD j n

ess.A/ d ; ƒess;".A/ C d " ; D¹ º D¹ 2 W j jD º that every dis.A/ is an accumulation point of .An/, n N, that no spectral 2 2 pollution occurs and that .An/n2N is "-pseudospectrally exact. 1084 S. Bögli

To this end, deﬁne

ak b2k1 T diag.Tk k N/; Tk ; D.T/ D.A/: WD W 2 WD 0 d WD k

Then it is easy to check that S A T is T -compact and the estimates in (4.1) WD are satisfied for all 0 C that are sufficiently far from .T/. The essential 2 spectrum ess.T/ consists of all accumulation points of .Tk/ ak; dk , k N, D¹ º 2 i.e. ess.T/ d . To find ƒess;".T/, note that, in the limit k , D¹ º !1

1 1 1 1 .ak / b2k1.ak / .dk / 1 .Tk / : k kD 0 .d /1 ! d k j j

This proves ƒess;".T/ C d " . Now the claims follow from D ¹ 2 W j j D º Theorem 4.1 and since ƒess;".T/ does not contain an open subset.

The following example is inﬂuenced by Shargorodsky’s example [31, Theo- rems 3.2, 3.3] of an operator with constant resolvent norm on an open set and whose matrix representation is block-diagonal. Here we perturb a block-diagonal operator with constant resolvent norm on an open set and arrive at an operator whose resolvent norm is also constant on an open set.

Example 4.3. Consider the neutral delay diﬀerential expression deﬁned by

.f /.t/ eit .f 00.t/ f 00.t // eit f.t/: WD C C C

For an extensive treatment of neutral diﬀerential equations with delay, see the monograph [1] (in particular Chapter 3 for second order equations). Let A be the realisation of in L2. ;/ with domain

0 2 f; f ACloc. ;/; f L . ;/; D.A/ f L2. ;/ 2 2 ; WD 2 W f. / f./; f 0. / f 0./ ² D D ³ where f is continued 2-periodically. With respect to the orthonormal basis ikt ek k Z D.A/ with ek.t/ e =p2, the operator A has the matrix ¹ W 2 º WD representation Local convergence of spectra and pseudospectra 1085

: : :: :: 0 1 T1 S1 B T S C 0 1 0 0 A B 0 0 C ;Tj ;Sj ; D B C D a 0 D 1 0 B : C j B T ::C B 1 C B C B :: C B :C B C @ A 2 aj 8j ; D t 2 2 D.A/ ..uj ;vj / /j 2Z .uj /j 2Z;.vj /j 2Z l .Z/; aj uj < : D W 2 j j 1 j 2Z ° X ±

We split A to T diag.Tj j Z/, D.T/ D.A/, and S A T . Note that S WD W 2 WD WD on D.S/ l2.Z/ is bounded and T -compact and S is T -compact, but T does D not have compact resolvent. Next we prove the existence of 0 %.T / such that 2 the estimates in (4.1) are satisﬁed. To this end, let 0 iR 0 and estimate 2 n¹ º

1 1 1 0 0 1 0 S.T 0/ sup Sj .Tj C1 0/ sup 2 C j 2 j k kD Z k kD Z a 0 1 I j 2 j 2 j 0 0 j j

1 2 one may check that also S .T 0/ .1 0 /= 0 . Hence (4.1) is k k C j j j j satisﬁed if 0 is suﬃciently large. j j Let Pn denote the orthogonal projection onto

Hn span ek k .2n/;:::;2n 1 ; WD ¹ W D º and let An and Tn denote the respective Galerkin approximations,

An PnA H ;Tn PnT H ; n N: WD j n WD j n 2

Note that det.An / det.Tn / for every C, which implies .An/ D 2 D .Tn/ paj j n;:::;n . Hence Theorem 4.1 ii) proves D ¹˙ W D º

.A/ paj j Z p8j j N0 : D ¹˙ W 2 ºD¹˙ W 2 º

Now we study the pseudospectra of A and T . 1086 S. Bögli

Figure 2. Eigenvalues (blue dots) and "-pseudospectra of the truncated 4n 4n matrices An for n 2 (top), n 4 (middle), n 6 (bottom) and " 1:5;1:4;:::;0:6;0:5. D D D D Local convergence of spectra and pseudospectra 1087

In Figure 2 the eigenvalues (blue dots) and nested "-pseudospectra (diﬀerent shades of grey) of An are shown for n 2; 4; 6 and " 0:5; 0:6; : : : ; 1:4; 1:5. D D As n is increased, for " > 1 the "-pseudospectra grow and seem to ﬁll the whole complex plane, whereas for " 1 they converge to ".A/ C. We prove these ¤ observations more rigorously. In fact, we show that there exists an open subset of the complex plane where the resolvent norms of A and T are constant (1=" D 1). So we cannot conclude "-pseudospectral exactness using Theorem 4.1 iii). However, "-pseudospectral inclusion follows from Theorem 3.6 i). In addition, 1 the upper block-triangular form of A implies that if x Hn, then Pn.A / x 1 1 21 D .An / xn. This yields .An / .A / and so k kk k

".An/ ".A/; n N: 2 Hence no "-pseudospectral pollution occurs. We calculate, for rei' with Re.2/ r2 cos.2'/ < 0, D D 1 1 2 .r max a ; 1 /2 .T /1 2 j : (4.4) j 2 2 4 C 2 ¹ º 2 k k D a aj r 2a r cos.2'/ a j j j j j C j jC

Hence, as in [4, Example 3.7], the resolvent norm of T is constant on a non-empty open set,

1 p5 1 .T rei'/1 1 if cos.2'/ < 0; r max C ; : k kD 2 cos.2'/ ° j j± One may check that

ess.T/ ess.T / ; .T K/ : D D ; C D ; K compact\ 1 In addition, since .Tj / 1, j , for any %.T /, we have k k ! ! 1 2 ƒess;".T/ ƒess;".T / , " 1. Therefore, using Theorems 3.8 i) and 4.1, D D ; ¤ C ; " 1; ess.A/ ess.A / ; ƒess;".A/ ƒess;".A / D D D ; D D ´ ; " 1: ; ¤ This implies, in particular, that .A /1 1 for all %.A/. Now we prove k k 2 that the resolvent norm is constant ( 1) on an open set. To this end, let ' be so 1 D that cos.2'/ < . We show that there exists r' > 0 such that 4 i' 1 .A re / 1; r r': (4.5) k kD 1088 S. Bögli

Deﬁne 1 1 ı1 ; ıj ; j Z 1 : WD 3 WD aj C1 cos.2'/ 2 n¹ º j j Then ıj 0 as j and ! j j!1 1 1 1 1 ıj 1aj ; j Z 0 ; ı sup ıj max ; < : D cos.2'/ 2 n¹ º WD j 2Z D 3 a1 cos.2'/ 2 j j ° j j±

Deﬁne functions fj Œ0; / R, j Z, by W 1 ! 2 2 4 r 1 f0.r/ r .1 ı/ C 2r WD ı1 and, for j 0, ¤ 4 fj .r/ r .1 ı/ 1 WD C 2 2 aj 1 aj r .1 2ı/ cos.2'/ sup cos.2'/ : C j j r j 2Zn¹1º aj C1 cos.2'/ j j j j

One may verify that there exists r' > 0 such that fj .r/ > 0 for all j Z and i' 2 r r'. We calculate, for re with r r', D 2 1 r2 1 S .T /1 2 ; j Z: j 1 j j j C2 2 4 2 C 2 k k D aj D r 2aj r cos.2'/ a 2 j j C j jC j We abbreviate

4 2 2 gj .r/ r 2aj r cos.2'/ a > 0; j Z: WD C j jC j 2 Then, with (4.4), we estimate for j 0, ¤

1 1 2 1 2 1 ıj 1 Sj 1.Tj / .Tj / ıj 1 k k k k 4 2 1 2 1 r .1 ıj / 1 aj r 2.1 ıj / cos.2'/ ıj aj C C j j ıj 1aj r ıj 1aj gj .r/ f .r/ j > 0; gj .r/ and analogously for j 0. So we arrive at D

1 1 2 1 2 1 ıj 1 Sj 1.Tj / > .Tj / ; j Z: ıj 1 k k k k 2 Local convergence of spectra and pseudospectra 1089

t 2 Let x .xj /j 2Z D.A/ D.T/ with xj .uj ;vj / C . Then D 2 D D 2 .A /x 2 k k .T /x Sx 2 Dk C k 2 .Tj /xj Sj xj C1 D k C k j 2Z X 2 1 2 .1 ıj / .Tj /xj 1 Sj xj C1 k k ıj k k j 2Z X 2 .1 ıj / .Tj /xj k k j 2Z X 1 1 2 2 1 Sj .Tj C1 / .Tj C1 /xj C1 ıj k k k k 1 1 2 2 1 ıj 1 Sj 1.Tj / .Tj /xj D ıj 1 k k k k j 2Z X 1 2 2 .Tj / .Tj /xj k k k k j 2Z X x 2; k k which implies .A /1 1 and hence (4.5). k k 4.2. Domain truncation of PDEs on Rd . In this application we study the sum of two partial differential operators in L2.Rd /, the first one of order k N and 2 the second one is of lower order and relatively compact. To this end, we use a t d multi-index ˛ .˛1;:::;˛d / R with ˛ ˛1 ˛d and D 2 j j WD CC j˛j ˛ d ˛ ˛1 ˛d t d D ; ; .1;:::;d / R : WD dx˛1 dx˛d WD 1 d D 2 1 d The differential expressions are of the form

1 ˛ 1 ˛ 1 2; 1 c˛D ; 2 b˛D ; WD C WD ij˛j WD ij˛j j˛Xjk j˛jXk1 where c˛ C. In order to reduce the technical diﬃculties, we assume that the 2 functions b˛ R C are suﬃciently smooth, W ! j˛j;1 d b˛ W .R /; ˛ k 1: 2 j j In addition, suppose that

ˇ lim D b˛.x/ 0; ˇ ˛ k 1: (4.6) jxj!1 D j j j j 1090 S. Bögli

d d Deﬁne the symbol p R C and principal symbol pk R C by W ! W ! ˛ ˛ p./ pk./ c˛ ; pk./ c˛ : WD C WD j˛jXk1 j˛XjDk We assume that p is elliptic, i.e.

d pk./ 0; R 0 : ¤ 2 n¹ º 2 d 2 d For n N let Pn be the orthogonal projection of L .R / onto L .. n;n/ /, 2 given by multiplication with the characteristic function .n;n/d . It is easy to see s that Pn I . ! 2 Theorem 4.4. Let A and An, n N, be realisations of in L .R/ and 2 L2.. n;n/d /, n N, respectively, with domains 2 D.A/ W k;2.Rd /; WD ˛ ˛ k;2 d D f ¹xj Dnº D f ¹xj Dnº; D.An/ f W .. n;n/ / j D j : WD ´ 2 W j d; ˛ k 1 µ j j gsr gsr i) We have An A and A A . ! n ! ii) The limiting essential spectra satisfy

d ess..An/n2N/ ess..A /n2N/ ess.A/ p./ R (4.7) D n D D¹ W 2 ºI

hence no spectral pollution occurs for the approximation .An/n2N of A, and every isolated dis.A/ is the limit of a sequence .n/n2N with n .An/, 2 2 n N. 2 iii) The limiting essential "-near spectra satisfy

ƒess;"..An/n2N/ ƒess;"..A /n2N/ D n ƒess;".A/ D (4.8) p./ z Rd ; z " D¹ C W 2 j jD º ".A/; and so .An/n2N is an "-pseudospectrally exact approximation of A.

2 Proof. Let T;S and Tn;Sn, n N, be the realisations of 1; 2 in L .R/ and 2 L2.. n;n/d /, n N, respectively, with domains 2

D.T/ D.S/ D.A/; D.Tn/ D.Sn/ D.An/; n N: D WD D WD 2 Local convergence of spectra and pseudospectra 1091

The operators T and Tn, n N, are normal; the symbol of the adjoint operators N 2 2 Rd T , Tn , n , is simply the complex conjugate symbol p. For f L . / 2 N N 2 2d denote its Fourier transform by fO and for n and fn L .. n;n/ / denote 2 d 2 2 by fn .fn.// Zd l .Z / the complex Fourier coeﬃcients, i.e. O D O 2 2 1 ix Rd f./ d f.x/e dx; ; O WD 2 Rd 2 .2/ Z 1 i n x Zd fn./ d f.x/e dx; : O WD 2 d 2 .2n/ Z.n;n/

Parseval’s identity yields that, if f D.T/, fn D.Tn/, 2 2 .T /f .p /f ; k kDk Ok .Tn /fn p fn k kD n O l2.Zd /I moreover, if p./ Rd , then …¹ W 2 º .T /1f .p /1f ; k kDk Ok 1 1 .Tn / fn p fn : k kD n O l2.Zd /

We readily conclude

d ess.T/ .T/ p./ R ; D D¹ W 2 º

".T/ C dist.;.T// < " ; D¹ 2 W º d ƒess;".T/ p./ z R ; z " ; D¹ C W 2 j jD º and, for n N, 2 d .Tn/ p Z ; D n W 2 ° ± ".Tn/ C dist.;.Tn//<" ; D¹ 2 W º and ﬁnally d ess..Tn/n2N/ p./ R ess.T /; ¹ W 2 ºD d ƒess;"..Tn/n2N/ p./ z R ; z " ƒess;".T /; ¹ C W 2 j jD ºD and the latter are equalities by Propositions 2.7 and 3.12. The same identities hold for the adjoint operators. This proves (4.7) and (4.8). 1092 S. Bögli

For any Rd we have 1 1 ˛ S.T / f b˛ L1./ sup p f ; (4.9a) k k k k 2Rd n k k j˛jk1 ˇ ˇ X ˇ ˇ ˇ 1ˇ 1 ˛ Sn.Tn / fn b˛ L1./ sup p fn : (4.9b) k k k k 2Rd n k k j˛jk1 ˇ ˇ X ˇ ˇ d ˇ1 1 ˇ By setting R , we see that S.T / , Sn.Tn / , n N, are D k k k k 2 uniformly bounded, and the uniform bound can be arbitrarily small by choosing far away from p./ Rd . The same argument also holds for the adjoint ¹ W 2 º operators. Let 0 be so that

1 1 S.T 0/ < 1; sup Sn.Tn 0/ < 1; (4.10a) k k n2N k k 1 1 S .T 0/ < 1; sup Sn .Tn 0/ < 1: (4.10b) k k n2N k k

Hence, in particular, S, Sn, S , Sn are respectively T -, Tn-, T -, Tn -bounded with relative bounds < 1 andso,by[22, Corollary 1], A T S and A T S . D C n D n C n By the assumptions (4.6) and [16, Theorem IX.8.2], the operator S is T - compact and S is T -compact. Hence [16, Theorem IX.2.1] implies

ess.A/ ess.T/; ess.A / ess.T / : D D Now we show that the assumptions (a)–(c) of Theorem 3.15 ii), iii) are satisﬁed for both A T S, An Tn Sn and A T S , An Tn D C D C D C D C Sn ; then the claims i)–iii) follow from the above arguments and together with Theorems 2.3, 3.6. We prove (c) before (b) as the proof of the latter relies on the former.

(a) The estimates are satisﬁed by the choice of 0, see (4.10). 1 d d (c) Let f C .R /, and let nf N be so large that supp f . nf ; nf / . 2 0 2 Then, for n nf , 1 1 .Tn 0/ Pnf Pn.T 0/ f; D 1 1 .T 0/ Pnf Pn.T 0/ f; n D 1 1 .Sn.Tn 0/ / Pnf .T 0/ S Pnf D n n 1 Pn.T 0/ S f D 1 Pn.S.T 0/ / f; D Local convergence of spectra and pseudospectra 1093

1 1 .S .T 0/ / Pnf .Tn 0/ SnPnf n n D 1 Pn.T 0/ Sf D 1 Pn.S .T 0/ / f: D Now the claimed strong convergences follow using (4.10) and the density of 1 Rd 2 Rd C0 . / in L . /. 1 (b) We prove that .Sn.Tn 0/ /n2N is a discretely compact sequence; for 1 N .Sn .Tn 0/ /n2N the argument is analogous. To this end, let I be an 2 d infinite subset and let fn L .. n;n/ /, n I , be a bounded sequence. Then 2 2 there exists an infinite subset I1 I such that .fn/n2I1 is weakly convergent in 2 d 1 L .R /; denote the weak limit by f . We show that Sn.Tn 0/ fn S.T 1 k 0/ f 0 as n I1, n . Assume that the claim is false, i.e. there exist k ! 2 ! 1 an infinite subset I2 I1 and ı > 0 so that 1 1 2 Sn.Tn 0/ fn S.T 0/ f ı; n I2: (4.11) k k 2 1 Note that (c) and Lemma 2.11 i) imply that .Sn.Tn / fn/n2I2 converges weakly 1 to S.T 0/ f . The assumption (4.6) yields

lim b˛ L1.Rd n.n;n/d / 0; ˛ k 1: n!1 k k D j j

Hence, by (4.9), there exists n0 N so large that 2 1 1 2 ı Rd d Sn.Tn / fn Rd d S.T / f < k n.n0;n0/ n.n0;n0/ k 2 for all n I2 with n n0; denote by I3 the set of all such n. An estimate similar 2 k;2 d to (4.9) reveals that the W .. n0; n0/ / norms 1 1 d .Tn 0/ fn d .T 0/ f k;2 d ; n I3; k .n0;n0/ .n0;n0/ kW ..n0;n0/ / 2 are uniformly bounded, and so the Sobolev embedding theorem yields

1 1 d Sn.Tn 0/ fn d S.T 0/ f 2 d 0: k .n0;n0/ .n0;n0/ kL ..n0;n0/ / ! Altogether we arrive at a contradiction to (4.11), which proves the claim.

It is convenient to represent An with respect to the Fourier basis and, in a further approximation step, to truncate the inﬁnite matrix to ﬁnite sections. We prove that these two approximation processes can be performed in one. For n N 2 let .n/ 1 i n x d Zd e .x/ d e ; x . n;n/ ; ; WD .2n/ 2 2 2 1094 S. Bögli denote the Fourier basis of L2.. n;n/d /. Note that the basis functions belong to 2 d D.An/. Let Qn denote the orthogonal projection of L .. n;n/ / onto

.n/ d span e Z ; 1 n : (4.12) ¹ W 2 k k º

s s One may check that Qn I and hence QnPn I . ! !

Theorem 4.5. The claims i)–iii) of Theorem 4.4 continue to hold if An is replaced by AnIn QnAn R.Q /. WD j n

Proof. Deﬁne TnIn QnTn R , n N. Note that QnTn TnInQn, n N. WD j .Qn/ 2 D 2 Analogously as in the proof of Theorem 4.4, we obtain

ess..TnIn/n2N/ ess.T/; ƒess;"..TnIn/n2N/ ƒess;".T/; D D and the respective equalities for the adjoint operators. It is easy to see that

SnIn QnSn R.Q /, n N, satisfy WD j n 2

1 1 SnIn.TnIn 0/ QnSn.Tn / R.Q /; n N; D j n 2

1 and hence the discrete compactness of .SnIn.TnIn 0/ /n2N follows from the 1 s one of .Sn.Tn 0/ /n2N and from Qn I . By an analogous reasoning, the 1 ! sequence .S .T 0/ /n2N is discretely compact. The rest of the proof nIn nIn follows the one of Theorem 4.4.

Example 4.6. Let d 1 and consider the constant-coeﬃcient diﬀerential opera- D tor d2 d T 2 ; D.T/ W 2;2.R/: WD dx2 dx WD

The above assumptions are satisﬁed if we perturb T by S b with a potential D b L1.R/ such that b.x/ 0 as x . For 2 j j! j j!1

2 b.x/ 20 sin.x/ex ; x R; WD 2 the numerically found eigenvalues and pseudospectra of the operator T S C truncated to the .2n 1/-dimensional subspace in (4.12) are shown in Figure 3. Local convergence of spectra and pseudospectra 1095

Figure 3. Eigenvalues (blue dots) and "-pseudospectra for " 23;22;:::;23 in interval D Œ 5; 10 Œ 7; 7 i of approximation AnIn for n 10 (left) and n 100 (right). C D D

The approximation is spectrally and "-pseudospectrally exact; the only discrete eigenvalue in the box Œ 5; 10 Œ 7; 7 i is 3:25. C Acknowledgements. The ﬁrst part of the paper is based on results of the au- thor’s Ph.D. thesis [2]. She would like to thank her doctoral advisor Christiane Tretter for the guidance. The work was supported by the Swiss National Science Foundation (SNF), grant no. 200020_146477 and Early Postdoc.Mobility project P2BEP2_159007.

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Received April 27, 2016; revised July 12, 2016

Sabine Bögli, Ludwig-Maximilians-Universität München, Theresienstr. 39, 80333 München, Germany e-mail: [email protected]