Enclosure of the Numerical Range of a Class of Non-Selfadjoint Rational Operator Functions

In this paper we introduce an enclosure of the numerical range of a class of rational operator functions. In contrast to the numerical range the presented enclosure can be computed exactly in the infinite dimensional case as well as in the finite dimensional case. Moreover, the new enclosure is minimal given only the numerical ranges of the operator coefficients and many characteristics of the numerical range can be obtained by investigating the enclosure. We introduce a pseudonumerical range and study an enclosure of this set. This enclosure provides a computable upper bound of the norm of the resolvent.


Introduction
The spectral properties of operator functions play an important role in mathematical analysis and in numerous of applications [Tre08,Lif89,APT02]. A classical tool to enclose the point spectrum is the numerical range and the closure of this set will under some conditions also enclose the essential spectrum but it is in most cases not possible to analytically determine the numerical range of an operator. Geometric properties of the numerical range of matrix polynomials and rational matrix functions were studied extensively [LR94,AMP02] and it is possible to numerically determine the shape of the numerical range of matrix polynomials [CNP02]. However, for high-dimensional matrix functions, for example generated by a discretization of a differential equation, the available algorithms are very time consuming.
In this paper we introduce an enclosure of the numerical range of a class of rational operator functions whose values are linear operators in a Hilbert space H. The new enclosure is applicable in the infinite dimensional case as well as in the finite dimensional case. Let A and B be selfadjoint operators in H, where B is non-zero and bounded. We consider rational operator functions of the form (1.1) T pωq :" A´ω 2´ω 1 numerical range can be computed exactly and is optimal given only the numerical ranges of A and of B. Resolvent estimates and pseudospectra are used to investigate quantitative properties of non-normal operators and operator functions [Dav07,TE05]. Estimates for the resolvent of general bounded analytic operator functions was considered in [MM01]. In order to derive a computable estimate for (1.1), we introduce a pseudonumerical range and study an enclosure of this set. The derived enclosure of the pseudonumerical range provides a computable upper bound of the norm of the resolvent in the complement of the new enclosure of the numerical range. The enclosure of the pseudospectra can be used to understand how the resolvent behaves outside the enclosure of the numerical range. Moreover, the enclosure of the pseudospectra shows where the resolvent potentially is large and can in the finite dimensional case be combined with a numerical estimate of the pseudospectra [TE05].
The paper is organized as follows: In Section 2, we present the enclosure of the numerical range, the theoretical framework used in the paper, and conditions for determining if ω P C belongs to the enclosure.
In Section 3, properties of the boundary of the enclosure is analyzed in detail. The presented results include location of local and global extreme points and determination whether a spectral gap in the form of a strip exists.
In Section 4, the ǫ-pseudonumerical range is introduced and we determine an enclosure of this set. Finally, the enclosures of the ǫ-pseudonumerical ranges is used to compute an upper estimate of the resolvent of (1.1).
Throughout this paper, we use the following notations. Let ω ℜ and ω ℑ denote, respectively, the real and imaginary parts of ω. If M is a subset of an Euclidean space, then BM denote the boundary of M. Let ?¨d enote the principal square root.

Enclosure of the numerical range
In this section we derive an enclosure of the numerical range of the operator function (1.1). Define for the non-negative real number c and positive d the constants (2.1) θ :" c c´d 2 4 , δ˘:"˘θ´i d 2 .
The numerical range of T can then be written in the form W pT q :" ď uPdom Azt0u tω P C : t pαu,βuq pωq " 0u.
For fixed values on the constants c, d the roots r n : RˆR Ñ C, n " 1, . . . , 4 of p u can be chosen to be continuous functions of the functionals pα u , β u q P R 2 . The numerical range of T can then be written as (2.4) W pT q " 4 ď n"1 ď uPdom Azt0u r n pα u , β u qztδ`, δ´u.
Let R :" R Y t˘8u denote the extended line of real numbers and set (2.5) Ω :" W pAqˆW pBq Ă RˆR.
Let C :" C Y t8u be the Riemann sphere and extend the functions r n , n " 1, . . . , 4 to r n : RˆR Ñ C such that the extension coincides with the limit values. The roots r n are given by particular pairs pα u , β u q P Ω and we obtain an enclosure W a pT q Ă C of the numerical range as (2.6) W pT q Ă W a pT q :" 4 ď n"1 r n pΩq, r n pΩq :" ď pα,βqPΩ r n pα, βq.
In the following we drop the index u and write p pα,βq pωq, since r n pα, βq in (2.6) are the roots of (2.3) independent of u.
The poles δ`and δ´are roots of p 2 and (2.7) is for large |α| a small perturbation of p 2 . Then, since the roots of a polynomial depend continuously on its coefficients, δ`and δ´are roots in the limit α Ñ˘8. There can be no other finite roots in the limit since the perturbation of p 2 is arbitrary small.
Proposition 2.2. The enclosure W a pT q has the following properties: i) W a pT q is symmetric with respect to the imaginary axis. ii) 0 P W a pT q if and only if 0 P W pAq or c " 0. iii) δ`P W a pT q if and only if W pAq is unbounded, 0 P W pBq or c " 0. iv) δ´P W a pT q if and only if W pAq is unbounded, or 0 P W pBq. v) 8 P W a pT q if and only if W pAq is unbounded.
Proof. i) The polynomial p pα,βq piωq has real coefficients. Hence, the symmetry follows from the complex conjugate root theorem. ii) Follows directly from (2.3), (2.6). iii) c " 0 implies δ`" 0 and δ`P W a pT q then follows from ii). The number p pα,βq pδ`q " βδ 2 is zero for β " 0, which implies δ`P W a pT q if 0 P W pBq. If W pAq is unbounded the statement follows directly from Lemma 2.1. Suppose none of the above holds, then p pα,βq pδ`q " βδ 2 ‰ 0, and since W pAq is bounded p pα,βq pωq ‰ 0 in a neighbourhood of δ`. The proof of iv) is similar to iii) with the difference δ´‰ 0 for c " 0. v) is immediate from Lemma 2.1.
Proof. Similar to Proposition 2.2.
The following propositions provide simple tests for ω P W a pT q.
Proposition 2.4. Assume ω " iω ℑ P iRzt0, δ`, δ´u. Then ω P W a pT q if and only if at least one of following conditions hold, Proof. From definition iω ℑ P W a pT q X iRzt0, δ`, δ´u if and only if there exists pα, βq P Ω such that Thus α is a non-constant real linear function in β. Since pα, βq P Ω and β belongs to a bounded set, r n pα 1 , β 1 q " iω ℑ for some pair pα 1 , β 1 q P BΩ. Equation (2.9) has two solutions unless the pair is a corner of Ω. Hence it is enough to investigate three of the line segments on BΩ to determine if iω ℑ P W a pT q. The converse holds trivially.
Corollary 2.7 iii) -iv) shows where in the complex plane W a pT q is located depending on the sign ofαpωq andβpωq as defined in Proposition 2.6.
The following Lemma and Lemma 2.15 show that γziR has finite number of self-intersecations.
Proof. i) Proposition 2.6 yields that ω P W a pT q if and only if pαpωq,βpωqq P Ω, thus only for pα, βq " pαpωq,βpωqq can r m pα, βq equal ω. Assume ω " r n pα, βq " r m pα, βq, n ‰ m. Then,´ω is also a double root since ω R iR and roots of p pα,βq are symmetric with respect to the imaginary axis. The result is then obtained using an ansatz with these two double roots. ii) Fix β P R and assume that ω " r n pα, βq is a root of p p¨,βq for α and for α 1 . Then 0 " pα´α 1 qpc´idω´ω 2 q and thus α " α 1 .
iii) The proof is similarly to ii).
Theorem 2.9. The set γ has the following properties: Let ω P W a pT q X iR, then the result follows from Corollary 2.3 and Proposition 2.4. ii) Assume δ`P W a pT qziR, then δ`P BW a pT q follows from Lemma 2.5 and Corollary 2.3 implies δ`P γ. The proof for δ´is similar and for 8 the result follows directly. Apart from iR Y tδ`, δ´, 8u, Lemma 2.8 i) yields that r n : RˆR Ñ C » R 2 is injective. Hence, a consequence of the invariance of domain theorem [Bro12] is that Br n pΩqziR " r n pBΩqziR, which implies BW a pT qz Ă γziR and ii) then follows from Lemma 2.8.
Corollary 2.10. The boundary of W a pT qziR is γziR.
Proposition 2.12. Assume c ą 0, then iµ P γ is an endpoint of a line segment of γ X iR if and only if iµ P R 1 9 YR 2 and mpiµq is odd. Further if iµ is an isolated point of γ X iR then iµ P R 1 9 YR 2 and mpiµq is even.
Proof. The result is first showed for iµ R t0, δ`, δ´,˘i8u. Assume iµ R N Y t0,˘i8u is an endpoint of a line segment or isolated point of γ X iR. Then for some pα, βq P BΩ, Assume that pα, βq R τ 1 , then since µ 2 c`dµ`µ 2 ą 0 it follows by similar arguments as given in Proposition 2.4 that there exist pα 1 , β 1 q P ΩzBΩ such that α 1`µ2μ 2 c`dµ`µ 2 β 1 " 0. Then, Lemma 2.8 ii)-iii) gives a contradiction. Hence pα, βq P τ 1 and iµ P R 1 . Assume iµ is a isolated point, then from the roots symmetry with respect to the imaginary axis it follows that mpiµq is even. Assume that iµ is endpoint of a line segment then from injectivity given by Lemma 2.8 ii)-iii) exactly one root must follow the line segment. Thus from the roots symmetry with respect to the imaginary axis it follows that mpiµq is odd.
If iµ P N zt0, δ`, δ´u similar argument gives the result for R 2 . For the converse assume iµ P R 1 and mpiµq odd. Then since µ 2 c`dµ`µ 2 ą 0 it follows that iµ is the root for an unique pair pα, βq P τ 1 . Assume that iµ is not the endpoint of a line segment, then since it is not an isolated point it must be inside a line segment. From injectivity given by Lemma 2.8 ii)-iii), symmetry with respect to the imaginary axis, and that mpiµq is odd, it follows that for pα 1 , β 1 q P BΩ sufficiently close to pα, βq there is exactly one simple root on the imaginary axis that is in the vicinity of iµ. Choose points iµ 1 , iµ 2 in the vicinity of iµ such that, µ 1 ă µ ă µ 2 and µ 2 i c`dµi`µ 2 i ą 0. Then there is some pα 1 , β 1 q, pα 2 , β 2 q P BΩ such that ą 0 there is a line of solutions pα, βq intersecting BΩ twice. Hence we can assume that α 1 " α 2 " α. By continuity there must exist a β 3 between β 1 and β 2 such that pα, β 3 q have the root iµ. But β 3 ‰ β which contradicts Lemma 2.8 iii). The proof for iµ P R 2 and mpiµq odd is similar. Assume iµ P t0, δ`, δ´u, then the result is shown by investing each case for iµ P R 1 9 YR 2 and when iµ is an endpoint of a line segment of γ X iR.
Remark 2.13. If c " 0, µ " 0 is always a solution to (2.3) and a similar results as in Proposition 2.12 can be obtained from the reduced cubic polynomial and by adding a single point at zero if it is not included in one of the other segments.
(2) Add an interval between iµ j , iµ j`1 in I if j is odd.
Proof. From Proposition 2.12 it follows that step 1 defines I as the set of endpoints of line segments of iµ P R 1 9 YR 2 , where iµ 1 is a local minimum. Then iµ 2 must be the endpoint of that segment, hence a local maximum. Doing this iteratively gives that for all odd j, iµ j is a local minimum and for even j, iµ j is a local maximum. Hence, step 2 sets I to γ X iR apart from isolated points. These are added in step 3.
The following Lemma and Lemma 2.8 implies that γziR has finite number of self-intersections.
Lemma 2.15. i) Fix α P Rzt0u, then p pα,¨q has a multiple root for at most 4 values β P R. ii) Fix β P Rzt0u, then p p¨,βq has a multiple root for at most 5 values α P R. iii) p p0,βq has a double root at 0 and the roots˘aβ`c´d 2 {4´id{2. iv) p pα,0q has the roots˘?α and˘ac´d 2 {4´id{2.
Proof. If α " 0 or both β " 0 and d " 2 ? c, then discriminant ∆ p pα,βq is zero and p pα,βq has a double root. For all other cases, we conclude from definition that ∆ p pα,βq is a fifth-degree polynomial in α and a fourth-degree polynomial in β.
Definition 2.16. Two disjoint sets Γ 1 , Γ 2 P C are neighbors if BΓ 1 X BΓ 2 contains at least one curve segment.
The algorithm presented in Proposition 2.17 is described in Figure 1.
Proposition 2.17. W a pT qziR " γziR if W pAq or W pBq constant. Otherwise W a pT qziR is obtained from γziR by the following algorithm: (1) Let O be the connected component of CzpγziRq containing values of ω with arbitrarily large imaginary parts.
Proof. If W pAq or W pBq are constant the result follows by definition. Corollary 2.7 shows that only one component of CzpγziRq contains values of ω with arbitrarily large imaginary parts. Hence the initial set O Ă CzW a pT qziR in Step 1 is well defined. From from Lemmata 2.8 and 2.15 it follows that γziR has a finite number of self-intersections and by definition γ has at most 4 components. Hence, CzpγziRq consist of finite number of components, which implies that the algorithm  Figure 1. Visualization of how W a pT qziR is obtained from γ using the algorithm presented in Proposition 2.17. Red and blue denotes points given by α P r0, 8s and α P r´8, 0q, respectively.
will terminate after finite number of steps. Corollary 2.10 yields that the set γziR is the boundary of a closed set and if two component of CzpγziRq are neighbours one is a subset of W a pT qziR and one is a subset of CzW a pT qziR. Thus if I and O are determined by the algorithm, I Ă W a pT qziR and O Ă CzW a pT qziR. Together with the termination criteria this yields the Proposition.
Remark 2.18. This problem is equivalent to 2-colorability of the dual graph of γziR.

Analysis of the enclosure of the numerical range
The functionT pλq :"´T p ? λq is analytic in the upper half-plane C`and ℑpT pλqu, uq ě 0, for λ P Cì f and only if W min pBq ě 0. Hence,T is a Nevanlinna function if and only if W min pBq ě 0. Since operator functions with applications in physics often are Nevanlinna functions, we analyze in this section the enclosure W a pT qziR under assumption W min pBq ě 0. However, the analysis when W min pBq is allowed to be negative is similar.
Let ω 1 , ω 2 , ω 3 , ω 4 be the roots of (2.3) and define From the relation between the coefficients and roots of a polynomial it follows that It is of interest to see when (2.3) has purely imaginary solutions and the following result shows that it depends on the sign of α. Figure 2. Examples of the set γ β P C, where red and blue denotes points given by α P r0, 8s and α P r´8, 0q, respectively.
i) If α ă 0, (2.3) has least two roots of the form iµ, µ P R, exactly one root where µ ą 0 and at least one root where c there are no pure imaginary root and if d ě 2 ? β`c there are at least two purely imaginary roots.
The set γ which determine the enclosure W a pT q, Theorem 2.9, depend on the rectangle BΩ. Hence there are two types of curves that are interesting to analyze: In subsection 3.1, β P W pBq is fixed and in subsection 3.2, α P W pAq is fixed. Properties of the roots of the polynomial (2.3) is then determined as a function of α and β, respectively.
3.1. Variation of the numerical range W pAq. In this section, we will in greater detail describe the subset of γ obtained when fixing β and vary α P W pAq. Define for β P W pBq the set γ β as Note that for ω P CzpiRY tδ`, δ´uq, ω P γ β is equivalent to β "βpωq in (2.11). The set γ β , can be parametrized in variable α P W pAq into a union of four curves. For β " 0, γ β is completely characterized by Lemma 2.15 and we will therefore assume β ą 0 in the rest of Section 3.1. Figure 2 illustrates possible behaviors of γ β .
If c " 0 then 0 R γ β ziR but all other statements hold as for the case c ą 0.
Proof. Assume iµ, µ P R, is a root of p pα,βq of order greater than one. Hence iµ has to be at least a double root and we set t 1 " 2µ, v 1 " µ 2 (3.1). The system (3.2) can then be written as . Solving (3.6) shows that µ is a solution if and only if µ " 0 or µ is a root of (3.5). Assume iµ P γ β ziR for some µ P R. Then, by symmetry with respect to the imaginary axis iµ is a double root of p pα,βq for some α. Hence µ " 0 or µ is a real solution to (3.5).
For the converse, assume that one of the poles is purely imaginary and that iµ P tδ 1 , δ 2 u is a root of q β , then from (3.5) it follows then that c P t0, d 2 {4u. For c " d 2 {4 is µ "´d{2 a solution to (3.5). Then in the limit α Ñ 8, δ 1 " δ 2 "´id{2 is a root of p pα,βq but p pα,βq do not have a purely imaginary root for any α P R`, which implies´iµ P γ β ziR. For c " 0 is µ " 0 a solution to (3.5), and thus a double root of p pα,βq for some α P R. Moreover, the origin is a root of p pα,βq for all α P R. Hence, the symmetry with respect to the imaginary axis implies that zero only can belong to the set γ β ziR if for some α, zero is a triple root of p pα,βq . An anzats with a triple root implies β " 0, which yields a contradiction. Now assume that µ " 0 or µ is a real solution to (3.5) and iµ R tδ 1 , δ 2 u. Then iµ is a double root of p pα,βq for some α P R and Lemma 2.8 ii) yields iµ P γ β ziR.
Let ∆ q β denote the discriminant of of q β .
Corollary 3.3. The polynomial (2.3) has a root iµ, µ P Rzt0u of multiplicity n ą 1 for some α P R if and only if µ is a root of (3.5) of multiplicity n´1.
Proof. The assumptions on the coefficients imply that q β can not have an quadruple root. A straightforward calculation shows that p p¨,βq has a quadruple root iµ for some α if and only if µ is a triple root of q β . Assume that p p¨,βq has a triple root iµ for some α. From (3.1) and (3.2) follows that this assumption is equivalent to ∆ q β " 0. Hence, q β has a multiple root and the multiplicity must be two. For n " 2, we showed that the multiplicity of a root of q β can not be larger then one. Then the result follows from Proposition 3.2.
The multiplicity of a real root µ of q β determine the number of segments of γ β ziR intersecting iµ, (if c " 0 there is no intersection in zero). Figure 3 c illustrates γ β for a case with a triple intersection of the imaginary axis. This can only happen when d " 2 ? β " 4 ? c and µ "´d{4, which implies that all sets γ β with this property are linear scalings of Figure 3 c.
For convenience, we set in Lemma 3.4 some constants to 8. These constants are used in Proposition 3.5.
Lemma 3.4. i) For β ă 4c, ∆ q β " 0 has an unique non-negative solution d 1 P p0, 2 ? cq. Figure 3. Sets γ β P C, where red denotes points given by α P r0, 8s and blue α P r´8, 0q. In a there is a double root of q β , (3.5), in b there are two distinct roots and in c there is one distinct root and one triple root.
ii) For 4c ď β ă 8c, ∆ q β " 0 has the three non-negative solutions If c " 0 we do not count the multiplicity of root 0. iv) The polynomial q β has zero real roots if d ă d 1 and four real roots if d 2 ď d ď d 3 , else it has two real roots.
Proof. Letd :" d 2 {4 and consider f pdq :" ∆ q β as a polynomial ind, where each positive root will correspond to exactly one positive solution d. By definition of the discriminant: For c " 0 the roots are 0, 27β{32 and the result follows. If c ą 0 existence of a rootd 1 P p0, cq follows from f p0q ą 0, f pcq ă 0. The discriminant of f is ∆ f " 2¨6 9 β 12 c 3 pβ´4cq 3 . i) Assume β ă 4c, then ∆ f ă 0 and thus f has only one real root. ii) Assume 4c ď β ă 8c, then ∆ f ě 0 and f is a cubic polynomial. It can be seen that f pβq ě 0 and f pdq Ñ´8,d Ñ 8. Hence there is one rootd 2 in pc, βs and one rootd 3 in rβ, 8q.
iii) Assume β ě 8c. Then f pβq ą 0, thus there is a rootd 2 in pc, βs. In the special case β " 8c, f is a quadratic polynomial and thus there are no more roots. Otherwise β ą 8c and then the last root will be negative.
iv) The sign of f will be negative if and only if d 1 ă d ă d 2 or d ą d 3 and thus in these cases q β have two roots, else it will either have zero or four roots. When d " 0, q β has no root and by continuity no roots for d ă d 1 . For d 2 ď d ď d 3 it holds that d ą 2 ? c and q β p´d{2´ad 2 {4´cq ă 0. Then, since the highest order term of q β is positive it must have at least one root and thus four roots.
Figure 3 a and c depicts γ β for d " d 1 and d " d 2 " d 3 , respectively.
Proposition 3.5. Let d k P R, k " 1, 2, 3 denote the constants defined in Lemma 3.4 and set For c ą 0, the set γ β ziR intersects the imaginary axis at zero and in the following points, counting multiplicity:

´d{2 is an intersection and there is one intersection in the interval
Proof. Assume c ą 0, then the intersections will coincide with roots of q β . i) From Lemma 3.4, iv) follows that there are no intersections when d P p0, d 1 q and two intersections for d P rd 1 , 2 ? cq. Since q β pµq ą 0 for µ ě´d{2, d P rd 1 , 2 ? cq the two intersections are in p´8,´d{2q.
ii) Follows from from straight forward computations.
iii) The value q β pµq is negative on p´8, d{2qzI 2 , thus there are no intersections in p´8, d{2qzI 2 . The function q β is convex on I 2 and q β p´8q ą 0. Hence, there is one intersection in I 2 . All other intersections are in p´d{2, 0q and the number of intersections is given by Lemma 3.4, iv). iv) qp´d{2q " 0, everything else follows by the same steps as iii). v) We have qp´d{2q ą 0, q 1 pµq ą 0 for µ P I 3 , and qp´pd`?d 2´4 cq{2q ă 0, thus one root of q is in I 3 . The function q β is convex on I 4 and q β p´8q ą 0. Hence, there are one intersection in I 4 . All other intersections are in p´d{2, 0q and the number of intersections is given by Lemma 3.4, iv).
Remark 3.6. Results similar to Proposition 3.5 hold also in the case c " 0 except the intersections at µ " 0 is false.
Proposition 3.7. ω P γ β ziR if and only if ω P t0, 8u (ω " 8 if c " 0) or ω ℑ ‰ 0 and one of the following four identities hold: Proof. From symmetry with respect to the imaginary axis we can choose t 1 " 2ω ℑ (3.1). The result then follows from straight forward computations using (3.2).
The system (3.2) can be solved for a given α by solving a fourth order polynomial to find ω. However, Proposition 3.7 shows that if ω ℑ is known, ω ℜ can be computed independently of α. This enables us to see γ β ziR as a multivalued function in ω ℑ .
Definition 3.8. A strip to a closed set Γ Ă C is defined as (3.10) S :" tω P C : where Γ X pR`is 0 q ‰ Ø and Γ X pR`is 1 q ‰ Ø. Then the set Γ X pR`is 0 q is called the local minimum points and Γ X pR`is 1 q is the local maximum points. The global minimum (maximum) is the point with the smallest (largest) imaginary part Γ. The extreme points of Γ is the union of the global and local maximum and minimum points. Corollary 3.9. The global minimum of γ β ziR is less than Impδ 2 q.
Proof. By simple computations it follows that the roots of f are (3.11). From Lemma 2.8 i) and Proposition 3.7 it follows that a double root µ of f exists if and only if µ "´d{4 "´?β{2 ă ? c. Then, Proposition 3.7 yields that the corresponding points on γ β are ω "˘ac´d{16´id{4. P 2 ω ℑ´Q ω ℑ is non-negative in an neighborhood of´d{4, thus it is not an extreme point but an self-intersection.
Assume that ω P γ β ziR is an extreme point and that f pω ℑ q ‰ 0. Then since one of the identities (3.8) holds it follows that P 2 ω ℑ´Q ω ℑ ą 0. Likewise P ω ℑȃ P 2 Hence, there exists an open interval I ω ℑ containing ω ℑ such that P µ˘b P 2 µ´Qµ´µ 2 ą 0 for each µ P I ω ℑ . Then (3.8) holds for any point in the interval, which contradict that ω is an extreme point. Hence it follows that ω ℑ is a distinct root of f . Now suppose ω ℑ is a distinct root of f then for every open interval I ω ℑ containing ω ℑ , there exist an µ P I ω ℑ such that f pµq ă 0, and it is thus an extreme point by identities (3.8).  Proof. From Proposition 3.2 follows that each intersection of the imaginary axis is equivalent to a root of q β as defined in (3.5). Assume iµ P γ β ziR is an extreme point.
Then by Proposition 3.7, P 2 µ´Qµ ě 0 and one of the identities 0 " P µ˘b P 2 µ´Qµμ 2 hold. If P 2 µ´Q µ " 0, then by continuity iµ is for some α a quadruple root of the polynomial p pα,βq defined in (2.3). Hence, by Proposition 3.7 the root can not be an extreme point. Furthermore, Corollary 3.3 implies that µ is a triple root of q β , thus not distinct. Assume P 2 µ´Qµ ą 0 and 0 ‰ P µ`b P 2 µ´Qµ´µ 2 . Then it follows from Lemma 3.10 that iµ P γ β ziR is not an extreme point. Hence, we have shown that 0 " P µ`b P 2 µ´Qµ´µ 2 and P 2 µ´Qµ ą 0. Assume µ is not distinct root of q β , then at least two segments of γ β ziR intersect iµ. Since P µ´b P 2 µ´Q µ´µ 2 ă 0, Proposition 3.7 implies that in some interval containing µ there is for a given ω ℑ at most two solutions ω. Combining these results shows that iµ is not an extreme point and the intersection must then be distinct. Assume P µ`b P 2 µ´Q µ´µ 2 " 0 and µ is a distinct root of q β . Then there is only one segment of γ β ziR intersecting iµ. Furthermore, P 2 µ´Q µ ą 0 and thus P µ´b P 2 µ´Q µ´µ 2 ă 0. Proposition 3.7 then implies that iµ is an extreme point.
Proof. If the bounded component of γ β ziR does not intersect iR it contains by continuity a root for all α P R. If the bounded component γ 1 Ă γ β ziR intersects with iR the curve is closed with an even number of intersections of the imaginary axis (counting multiplicity in (3.5)). Assume that there is an α P R such that no root is on γ 1 or purely imaginary with imaginary part in J. Then, by definition (3.4), r n pα, βq is not on γ 1 Y iJ for n " 1, 2, 3, 4. From the continuity of the roots it follows that there exist an α 1 such that r n pα 1 , βq P γ 1 Y iJ and for a sufficiently small |ǫ|, r n pα 1`ǫ , βq, n " 1, 2, 3, 4 is not on γ 1 Y iJ. Since this can only happen on the imaginary axis it follows that (2.3) for α 1 has a purely imaginary multiple root iµ on γ 1 . Thus for some ordering of the roots r 1 pα 1 , βq " r 2 pα 1 , βq " iµ.
Since r 1 pα 1`ǫ , βq R γ 1 and r 2 pα 1`ǫ , βq R γ 1 it follows by continuity that they are imaginary. By Lemma 2.8 ii) one of the roots has larger imaginary part and one smaller imaginary part than µ. Then if both are outside J, it follows that J will consist of only one point. Hence µ is a double root of (3.5) and from Corollary 3.3, iµ is a triple root of p pα,βq . Hence r 3 pα 1 , βq " iµ and by injectivity (Lemma 2.8) one root will belong to γ 1 for α 1`ǫ , a contradiction.
If 2 ? β ě d ą d 2 , it follows that (3.5) has four distinct real roots (if d 3 " 2 ? β it follows from Lemma 3.4 that d 2 " 2 ? β and this contradicts 2 ? β ě d ą d 2 ). Thus from Corollary 3.3 there are five intersections of γ β ziR with the imaginary axis. Assume that the global minimum is not on the imaginary axis. Then it follows from Lemma 3.10 and d 2 ą 2 ? c that the global minimum is larger than´d{2, which contradicts Corollary 3.9. Hence the minimum is on the imaginary axis and thus given by the root µ 1 , where P µ1`b P 2 µ1´Qµ1´µ 2 1 " 0 by Lemma 3.11. Assume that there is no strip. Then Lemma 3.11 implies P µ`b P 2 µ´Qµ´µ 2 ą 0, with P 2 µ´Q µ ě 0 for all µ P pµ 1 , 0q. Define for µ P pµ 1 , 0q the function f pµq :" P µ´b P 2 µ´Qµ´µ 2 . Then Proposition 3.7 implies f pµ i q " 0 for i " 2, 3, 4. Take i P t2, 3, 4u and assume that f pµq is either positive or negative in an open punctured interval around µ i . Then it follows from Corollary 3.3 that µ i is not a distinct root of q β . Hence f pµq alternates signs between the roots. By Proposition 3.7 there must thus be two bounded components of γ β ziR. Since d ă 2 ? β`c, Proposition 3.5 implies µ i ą´d{2 for i " 2, 3, 4. Then from Lemma 3.13 it follows that both poles are larger than´d{2, which gives a contradiction and a strip must therefore exist.
Assume that there are at least two strips, then there must be at least three components of γ β ziR, one is unbounded and two bounded. Lemma 3.13 implies that the bounded components will both enclose a pole. This yields that the poles are imaginary and thus each bounded component intersects the imaginary axis twice and we have four real roots of (3.5). Since the imaginary parts of these roots approach 0 as the real parts approach˘8, the points 0 and 8 will be in the same component with no other intersections of the imaginary axis. This means that for α ě 0 there are two roots in the unbounded component. By Lemma 3.13 there are always at least one root on or enclosed by a bounded component. Hence, for α ě 0 there is only one root in each bounded component. Thus due to symmetry the roots in the bounded components are imaginary for all α ě 0. Hence, if ω belongs to a bounded component of γ β ziR then ω ℑ ď´d{2 and thus all the solutions µ to (3.5) satisfies µ ď´d{2, which contradicts Proposition 3.5.
If d ď minp2 ? β, d 2 q, it follows that (3.5) has at most 2 distinct roots, µ 1 , µ 2 . Hence, Lemma 3.10 implies that there can only be one extreme point not on the imaginary axis and only if d ă 2 ? c. Assume a strip exists, then d ă 2 ? c must hold. By Lemma 3.10 and Proposition 3.5 all possible extreme points have imaginary part smaller than´d{2, which is the imaginary part of the poles. Hence by Lemma 3.13 there is no strip. c then the local minimum point is iµ 3 . Else the local minimum points are not on the imaginary axis and s 1 " p´d`ad 2´4 βq{4.
Proof. First show that s 0 " µ 2 if the maximum point is on the imaginary axis and s 0 " p´d´ad 2´4 βq{4 if the maximum is not on the imaginary axis. If a strip exists then it follows from Corollary 2.7 that there is exactly one unbounded component and one bounded component of γ β ziR. By continuity the bounded component intersects the imaginary axis an even number of times. If the local maximum is on the imaginary axis it must thus be the largest root of the bounded component. Thus the root denoted µ 2 in Definition 3.14 if the bounded component intersects the imaginary axis two times and the root µ 4 if there are four intersections. Assume there are four intersections with the imaginary axis. This leads to a contradiction by arguments analogous to the proof of the uniqueness of a strip in Proposition 3.15. Thus, the local maximum is iµ 2 , similarly it follows that the local minimum is iµ 3 .
Assume that the local maximum is not on the imaginary axis. Then f ps 0 q " s 2 0 pP 2 s0´Qs0 q " 0 from Lemma 3.10 and s 0 satisfies one of (3.11), i) or ii). Since it is a local maximum, for sufficiently small ǫ ą 0, f ps 0´ǫ q ą 0 and f ps 0`ǫ q ă 0, which implies s 0 " p´d`d a 1`4β{p4c´d 2 qq{4 or s 0 " p´d´ad 2´4 βq{4. Assume that s 0 " p´d`d a 1`4β{p4c´d 2 qq{4, then d 2 ą 4β`4c since otherwise s 0 is not negative. Then f ps 0 q :" P s0˘a P 2 c . For the local minimum the same idea is used.
ii) Assume β ě 4c then by Lemma 3.4, d 2 ď 2 ? β. Hence Proposition 3.15 implies that there is a gap if and only if d ą d 2 . The point ω is a local maximum not on the imaginary axis if and only if ω ℑ " p´d´ad 2´4 βq{4 and P ω ℑ´ω 2 ℑ ą 0, which never holds. For the local minimum the same idea is used for ω ℑ " p´d`ad 2´4 βq{4 and then it follows that P ω ℑ´ω  Figure 4. The set γ β P C, where red denotes points given by α P r0, 8s and blue α P r´8, 0q. The figure describes how γ β , changes with increasing d. In picture a there is no strip. In picture b there is a strip and the minimum is moving to the imaginary axis.
In c the local maximum is moved to the imaginary axis. Proof. From Proposition 3.5, Lemma 3.10, and Corollary 3.9 it follows that ω with imaginary part (3.14) and iµ 1 are the only possible global minimums. Moreover if the minimum is not on the imaginary axis then d ă 2 ? c since otherwise ω ℑ ă Impδ 2 q does not hold. It thus follows that if c " 0 the minimum is iµ. Assume c ą 0, then ω is a minimum not on the imaginary axis if and only if d ă 2 ? c and by Proposition 3.7, P ω ℑ´ω 2 ℑ ą 0. This holds if and only if d ă 2 ad , whered is the unique solution to (3.13) satisfying 0 ăd ă c. Else the minimum must be on the imaginary axis and thus iµ 1 .
In Figure 4 b the local minimum is clearly not on the imaginary axis but by increasing d, Figure 4 c is obtained, where the local maximum point is on the imaginary axis.
In the following we study the dependence of γ β ziR on the parameter d and denote for fixed d ą 0 this set by γ β pdqziR.
Proposition 3.18. The extreme points of γ β pdqziR are continuous in d and the global minimum is decreasing with d. Furthermore, if a strip S as defined in (3.10) exists, then s 0 is strictly decreasing and s 1 is strictly increasing with respect to d.
Proof. Propositions 3.16, 3.17, and 3.7 yields the continuity of the extreme points. Hence, it is enough to show the results for the extreme points on and off the imaginary axis separately. For each extreme point on γ β ziR, the result follows directly from Propositions 3.16 and 3.17. All other extreme points can be written in the form ω " iµ, µ P R where µ is a solution to (3.5) and the minimum is then µ 1 as defined in (3.12). Let q d β pµq denote the polynomial (3.5) for a given d ą 0 and take ǫ ą 0. Then Let µ d i for i " 1, 2, 3, 4 denote the real roots of q d β ordered non-decreasingly. If the minimum is on the imaginary axis then it is given by µ d 1 and Proposition 3.5 implies pµ d 1 q 2`d µ d 1`c`β {4 ą 0. Thus for ǫ ą 0 small enough q d`ǫ β pµ d 1 q ă 0 and (3.15) implies that the minimum of γ β pdqziR is decreasing in d.
Assume a strip S exists and that the local maximum (minimum) point is 0 (is 1 ) is on the imaginary axis. Then Proposition 3.16 implies s 0 " µ d 2 (s 1 " µ d 3 ) and we conclude that q d β pµq ą 0 for µ P S. In particular the maximum (minimum) is decreasing (inreasing) if and only if for ǫ ą 0 small enough q d`ǫ β pµ d 2 q ą 0 pq d`ǫ β pµ d 3 q ą 0q. In general, q d`ǫ β pµq ą 0 for ǫ ą 0 small enough if and only if µ 2`d µ`c`β{4 ď 0, which is equivalent to Hence, what remains is to show that all local extreme points µ d 2 , µ d 3 are in I d . It can be seen that if µd 2 , µd 3 P Id, then µ d 2 , µ d 3 P I d for all d ěd. If β ă 4c then by Proposition 3.16 i), only the local maximum can be on the imaginary axis. Moreover, the condition d ě d 0 :" pβ`4cq{p2 ? cq holds, which together with µ d0 2 "´?c P I d0 yields the result. If β ą 4c, there is a strip if and only if d ą d 2 , further µ d2 2 " µ d2 3 . Thus it is enough to show µ d2 2 P I d2 to prove the clame. Since β ą 4c, it follows from Lemma 3.4 that d 2 ď 2 ? β ă d 0 . Hence, for d 0´d2 ě ǫ ą 0 the set γ β pd 2`ǫ qziR has both the local minimum and maximum on the the imaginary axis. But then for some d P pd 2 , d 0 q either µ d 2 is decreasing or µ d 3 is increasing. Hence either µ d 2 P I d or µ d 3 P I d . Since this holds for arbitrarily small ǫ ą 0, µ d2 2 " µ d2 3 P I d2 . For β " 4c, the result follows immediately since the roots of (3.5) are continuous in β and Proposition 3.16.
For α " 0 this is completely solved in Lemma 2.15, therefore assume that α P Rzt0u.
Remark 3.19. In the definition of γ α , we set W pBq " r0, 8s since r0, 8s is the smallest closed interval containing W pBq for all bounded B. The limit of the roots r n pα, βq are 0 and˘8´id{2 as β Ñ 8. These the points are in γ α but for α ‰ 0 not in γ α,W pBq .
For ω P CzpiR Y tδ`, δ´uq, ω P γ β is equivalent to ω P Π β as defined in (2.15), and α "αpωq in equation (2.12). The following results for this curve are similar to  Figure 5. Examples of the set γ α P C that visualize how the set depends on the sign of α. In picture a, α ą 0, in picture b, α " 0 and in picture c, α ă 0.
the results for γ β , but the behavior will greatly depend on the sign of α as can be seen in Proposition 2.7 and in Figure 5.
Proof. The proof follows the same steps as in Proposition 3.2, with the additional condition β ě 0, which by Corollary 2.7 iii) simplifies to ν ď´2 c d or ν " 0. Corollary 3.21. The polynomial (2.3) has a root iν, ν P Rzt0u of multiplicity n ą 1 for some β P r0, 8q if and only if ν ď´2c{d and ν is a root of (3.17) of multiplicity n´1.
Proof. Similar to Corollary 3.3, with the additional condition β ě 0.
For c " 0 the roots are 0, 27α{4 and the result follows. If c ą 0 the discriminant of f is ∆ f "´2¨6 9 α 9 c 3 pα´cq 4 pα`cq 2 . i) Assume α ą 0 then ∆ f ď 0 with equality only if c " α. In that cased " 4c is a triple root of f , else ∆ f ă 0 and then there is one real root. Hence, there is in each case exactly one real solution to f pdq " 0. Denote this solutiond 0 and define d 0 " 2 ad 0 . Then d 0 is the unique positive solution to ∆ qa " 0. Further since f p4αq, f p4cq ď 0 and f pdq Ñ 8,d Ñ 8 it follows that the unique positive solution d 0 P r4 maxpα, cq, 8q. Ifd ăd 0 then f pdq ă 0 and thus there are two real roots of q α . Ifd ěd 0 then f pdq ď 0 and since q α p0q ď 0 there is always one root and thus it must be four. ii) Assume α ă 0, then there are always three real roots since ∆ f ě 0. From f p´8cq ď 0, f p0q ą 0 and f pdq Ñ 8,d Ñ´8, exactly one is positive. Since f pcq ď 0 the unique solutiond is in p0, cs. Assume d ă d 0 , then by continuity there is no roots since q α has no real root for d " 0 and ∆ qα ą 0 for d ă d 0 . Assumê d ěd 0 , then f pdq ď 0 and thus there is two roots of q α . ? c, the point´2c{d "´d{2 is a simple root of q α and by Proposition 3.20, there is a simple intersection of the imaginary axis. For ν ă´d{2, q α pνq ą 0 and hence no more intersections of the imaginary axis. For d ą 2 ? c, q α p´2c{dq ă 0, thus there is either one or three solutions of (3.17). If 2 ? c ă d ă d 0 , there is one intersection. Assume d ě d 0 . Then q α pνq ă 0 for ν P p´2c{d, 0q and q α is convex for ν ě 0. Hence, (3.17) has three solutions. Now assume c " 0, then Proposition 3.20 shows that zero is a false root and thus in this case there is one less intersection of the imaginary axis.
ii), For ν ď´2c{d the derivative q 1 α pνq is non-positive with equality only if α " c, thus there is at most one solution to (3.17). If 2 ? c ą d then q α p´2c{dq ą 0 and there is no solution. For 2 ? c ď d, q α p´2c{dq ď 0, hence from Proposition 3.20 and α ě´c it follows that (3.17) has one solution. If 2 ? c " d and α "´c, theń 2c{d is a double root of (3.5) and by Corollary 3.21 a triple root of p pα,0q . From the injectivity stated in Lemma 2.8 iii) follows that there is only one intersection of the imaginary axis at this point. The result then follows from Proposition 3.20.
iii) If d ă d 0 then there are no real solutions to (3.17). If d 0 ď d ă 2 ? c, then q α pνq ą 0 for ν ą´d{2. For´d{2 ě ν ą´2c{d we have Thus, the two roots satisfy ν ď´2c{d. Assume d " 2 ? c, then by Proposition 3.17 the value´2c{d is not an intersection of the imaginary axis. However, there is an intersection at the distinct root 3 a αd{2, which shows that we have one root. For ? c ą 0 it follows that q α p´2c{dq ă 0 and it can be seen that (3.17) has one solution.
The multiplicity of a real root ν ă´2c{d of q α in (3.17) will determine the number of segments of γ α ziR intersecting iν. However, this will in general not hold if´2c{d is a root of q α . This case is however addressed in Proposition 3.23. Figure 6 b, shows an example where´2c{d " 2 is an intersection of the imaginary axis, while in Figure 6 c,´2c{d " 2 is not an intersection despite being a root of q α . Cases with different number of intersections are illustrated in Figure 5 and in Figure 6.
Proposition 3.24. Assume c ą 0 then ω P γ α ziR if and only if ω " 8´d{2 (in the limit ω ℑ "´d{2), ω " δ˘(only if d ď 2 ? c which is strict if α ă´c) or ω P Π β ztR´id{2u and one of the following four identities hold: If c " 0 then 0 R γ α ziR but all other statements hold as for the case c ą 0.
Remark 3.25. Since in the limit β Ñ 8 the roots r n pα, βq approaches 0 and˘8í d{2, it is convenient to assume the imaginary part in infinity is´d{2. Using this convention, for α ą 0 the global maximum is 0 and the global minimum is´d{2. For α ă 0 the global maximum is´d{2.
Lemma 3.26. A point ω P γ α ziR, with ω ℑ R t0,´d{2u is an extreme point in the sense of Definition 3.8 if and only if it is a distinct root to gpω ℑ q :" pd2 ω ℑ q 2 pR 2 ω ℑ´S ω ℑ q. The roots of g are where ω "˘ac´d 2 {16´id{2. Then, ω is a self-intersection of γ α ziR.
Proof. The proof is similar to that of Lemma 3.10.
An example of a self-intersection of γ α ziR can be seen in Figure 6 b.
Lemma 3.27. A point iν P iR where ν P Rzt´d{2, 0u is an extreme point to γ α ziR if and only if 0 " R ν`a R 2 ν´Sν´ν 2 and ν is a distinct intersection of the imaginary axis.
Lemma 3.28. For any β P r0, 8q each bounded closed curve component of γ α ziR either contains or encloses at least one root to (2.3).
Proof. The proof is similar to the proof of Lemma 3.28 with the exception that any bounded component is not necessarily closed. Proof. Similar to the proof of Proposition 3.15, using Proposition 3.24 and Lemmata 3.26, 3.27 and 3.28.
In the following we study the dependence of γ α ziR on the parameter d and denote for fixed d ą 0 this set by γ α pdqziR. The mapping d Þ Ñ γ α pdq is a continuous deformation since γ α is given by roots of polynomials. Moreover, by Lemma 3.28 and that the number of roots of (3.17) is finite it follows that γ α pdqziR is a continuous deformation, except in the case d " 2 ? c and α ă´c at ν "´d{2. (3.23) i) holds and ω ℑ " p´d´d a 1`α{cq{4 . Else the global minimum point is iν 1 . If c " 0 the global minimum point is always iν 1 .
Proof. The proof is similar to Proposition 3.16.
Proposition 3.32. The extreme points are continuous in d.
i) If α ą c and a strip S (3.10) exists, then the local minimum of γ α ziR is increasing in d, and the local maximum is decreasing in d. ii) If c " α ą 0 and a strip S exists, then the local minimum of γ α ziR is´i ? c, and the local maximum is´id{4´i a d 2 {16´c. iii) If c ą α ą 0 then the local extreme points of γ α ziR is decreasing in d. iv) For α ă 0 the global minimum of γ α ziR is decreasing in d.
Proof. The continuity in d of the extreme points follows from Propositions 3.31 and 3.24.
i)-iii) Assume that a strip S exists. If an extreme point is located not on the imaginary axis then α ă c by Lemma 3.26 and then the result follows from (3.21). Thus it sufficient to show the result for the roots of (3.17) and for the case α " c it follows directly. Hence, assume that α ‰ c. Similarly as in Proposition 3.18, observe the polynomial shifted by some small ǫ ą 0 is For α ą 0 with a strip it follows from Proposition 3.29 that d ě d 0 since otherwise the minimum and maximum is not on the imaginary axis, and thus there are three roots. The local maximum is iν 1 and the local minimum is iν 2 by Proposition 3.31. The local maximum (minimum) is decreasing (increasing) if and only if, q d`ǫ α pν d 1 q ă 0 pq d`ǫ α pν d 2 q ă 0q, for ǫ ą 0 sufficiently small. In general, if q d α pνq " 0, then q d`ǫ α pνq ą 0 for ǫ ą 0 small enough if and only if pν 2´α q ą 0, and thus ν ă´?α. Hence, what to be shown is wether ν d 1 , ν d 2 ă´?α. Since q α p´?αq " αpα´cq, there is an even number of roots in p´8,´?αs if α ą c and an odd number of roots p´8,´?αs if α ă c. The polynomial q d α pνq is concave only on the interval r´d{4, 0s, hence there can at most be two roots in this interval. From Lemma 3.22 and that d ě d 0 it follows that´d{4 ď´?α and thus there has to be at least one root in the interval p´8,´d{4s. Hence, if α ą c there are two roots in p´8,´?αs, thus ν d 1 , ν d 2 ă´?α, this proves i). If α ă c it follows that q 1 α p´?αq and q 1 α p´d{4q are positive and since q α is concave on r´d{4,´?αs there is at most one root in that interval. On p´8,´d{4s, q α is convex and since q α p´d{4q ă 0 it follows that q α has exactly one root in this interval. Since the number of roots is odd in p´8,´?αs it follows that there is only one root in the interval, this proves iii).
iv) The result follows from Lemma 3.26 if the minimum is not on the imaginary axis. Assume α ă 0 and that the minimum is on the imaginary axis. Then the minimum will be iν d 1 , which implies ppν d 1 q 2´α q ą 0 and thus the minimum is decreasing, due to (3.24).

Resolvent approximation
In this section, we assume ǫ ą 0. The ǫ-pseudospectrum σ ǫ pT q is the union of σpT q and the set of all ω P C such that }T´1pωq} ą ǫ´1. An equivalent condition for ω P σ ǫ pT q is that there exist a function u P dom T , }u} " 1 for which }T pωqu} ă ǫ. Such u is called an approximate eigenvector or ǫ-pseudomode [Dav07,p. 255]. To be able to see how well-behaved T´1 is close to W a pT q as defined in (2.6) we will make an upper estimate of the resolvent for the rational function (1.1).
Remark 4.2. For ω P tδ 1 , δ 2 , 8u, ω P W a pT q if and only if ω P W ǫ a pT q for all ǫ ą 0. Figure 7 illustrates W a pT q and W ǫ a pT q in two cases. Note in particular that the distance between a point ω P BW ǫ a pT q and BW a pT q is not constant. Define the set, (4.2) γ ǫ :" BΓ ǫ zW a pT q, Γ ǫ :" tω P Czγ : Dpα, βq P BΩ, |tpωq| ă ǫu.
Remark 4.4. From the minimum modulus principle it follows that W ǫ a pT q has no new components compared with W a pT q. However, components disconnected in W a pT q might be connected in W ǫ a pT q; see Figure 7 a.
Proof. ω P W ǫ a pT q if and only if ǫ ą min pα,βq |t pα,βq pωq| ": ǫ 0 . In the sense of (4.4), this is a constrained linear least squares optimization problem. If κ ℜ " 0 the result is trivial. Otherwise, since the problem is linear the minimizing value α is either on one of the endpoint of W pAq or an α that makes the second equation solvable in β P W pBq. Computing the optimal β P W pBq in all of these cases gives three possible pairs in Ω that minimizes (4.4), the result follows.
Proposition 4.7. Let A ǫ be an arbitrary selfadjoint operator with W pA ǫ q " rW min pAqǫ , W max pAq`ǫs and define T ǫ pωq :" A ǫ´ω2´ω 2 c´idω´ω 2 B Figure 8. In the figure W pAq " r´32, 4s, W pBq " r0, 4s, c " 4 and d " 4. For ω P CzpW a pT q Y tδ`, δ´uq the upper bound of }T´1pωq} from Corollary 4.6 is visualized. For ω P W a pT q the resolvent is computed numerically for particular matrices A and B.
Then W ǫ a pT q X iR " W a pT ǫ q X iR. Proof. Assume ω P iRztδ`, δ´u, then as in proof of Theorem 4.3 we have ω P W ǫ a pT q if and only if, min pα,βqPΩ |t pα,βq pωq| ď ǫ. Hence, ω P W ǫ a pT q if and only if (4.5) α`ω 2 ℑ`ω 2 ℑ c`dω ℑ`ω 2 ℑ β " e for some pα, βq P Ω and some e with |e| ď ǫ. Assume ω P iRztδ`, δ´u satisfies (4.5), then α´e P R, α´e P W pA ǫ q and thus ω P W a pT ǫ q. The converse is obvious. Hence, W ǫ a pT q X iRztδ`, δ 1 u " W a pT ǫ q X iRztδ`, δ 1 u, which by continuity yields W ǫ a pT q X iR " W a pT ǫ q X iR. The set W ǫ a pT q X iR can then be obtained by the algorithm given in Proposition 2.14.