J. Numer. Anal. Approx. Theory, vol. 44 (2015) no. 2, pp. 127–145 ictp.acad.ro/jnaat
MULTICENTRIC CALCULUS AND THE RIESZ PROJECTION
DIANA APETREI∗and OLAVI NEVANLINNA∗
Abstract. In multicentric holomorphic calculus one represents the functionϕ using a new polynomial variablew=p(z) in such a way that when it is evaluated at the operatorA,thenp(A) is small in norm. Usually it is assumed thatphas distinct roots. In this paper we discuss two related problems, the separation of a compact set (such as the spectrum) into different components by a polynomial lemniscate, respectively the application of the Calculus to the computation and the estimation of the Riesz spectral projection. It may then become desirable the use ofp(z)nas a new variable. We also develop the necessary modifications to incorporate the multiplicities in the roots.
MSC 2010. 30B99, 32E30, 34L16, 47A10, 47A60.
Keywords. multicentric calculus, Riesz projections, spectral projections, sign function of an operator, lemniscates.
1. INTRODUCTION
Let p(z) be a polynomial of degree d with distinct roots λ1, . . . , λd. In multicentric holomorphic calculus the polynomial is taken as a new variable w=p(z) and functions ϕ(z) are represented with the help of a vector-valued function f, mapping w 7→ f(w) ∈ Cd, [9]. For example, sets bounded by lemniscates |p(z)|=ρ are then mapped onto discs |w| ≤ρ and forρ small, f has a rapidly converging Taylor series.
The multicentric representation then yields a functional calculus for oper- ators (or matrices) A, if one has found a polynomial p such that kp(A)k is small. In fact, denote
Vp(A) =z∈C : |p(z)| ≤ kp(A)k
and observe that, by spectral mapping theorem, the spectrum σ(A) satisfies σ(A) ⊂Vp(A). If f has a rapidly converging Taylor series for |w| ≤ kp(A)k, thenϕ(A) can be written down by a rapidly converging explicit series expan- sion.
SinceVp(A) can have several components, one can defineϕ= 1 in the neigh- borhood of some components, while ϕ = 0 in a neighborhood of the others.
∗Aalto University, Department of Mathematics and Systems Analysis, P.O.Box 11100, Otakaari 1M, Espoo, FI-00076 Aalto, Finland, email: {Diana.Apetrei, Olavi.Nevanlinna}@[email protected].
Thenϕ(A) represents the spectral projection onto the invariant subspace cor- responding to the part of the spectrum where ϕ= 1. The spectral projection satisfies
ϕ(A) = 2πi1 Z
γ
(λI−A)−1dλ,
whereγ surrounds the appropriate components of the spectrum, but the com- putational approach does not need the evaluation of the contour integral. The coefficients for the Taylor series off can be computed with explicit recursion from those of ϕat the local centers λj [9]. The approach also yields a bound forkϕ(A)kby a generalization of the von Neumann theorem for contractions, see [7]. If the scalar functionϕis not holomorphic at the spectrum, the multi- centric calculus leads to a new functional calculus to deal with, e.g. nontrivial Jordan blocks [10].
In this paper we take a closer look at the computation of spectral projections in finding the stable and unstable invariant subspaces of an operator A. The direct approach would be to ask for a polynomial p with distinct roots such that
Vp(A)∩iR=∅
and then apply the calculus with ϕ = 1 for Re z > 0, respectively ϕ = 0 for Rez < 0. However, we discuss this as two different subjects, one being the separation of the spectrum and the other being the computation of the projection.
In order to discuss the separation, we denote V(p, ρ) ={z∈C :|p(z)| ≤ρ}
and we letK =σ(A),then we ask what is the minimal degree of a polynomial such that
(1.1) K ⊂V(p, ρ) andV(p, ρ)∩iR=∅.
holds. By Hilbert’s lemniscate theorem, see e.g. [12], such a polynomial with a minimal degree always exists. We model this question by considering in place of σ(A) two lines parallel to the imaginary axis as follows
K =z=x+iy : x∈ {−1,1}, |y| ≤tan(α)
and derive a sample of polynomials for which (1.1) holds when α grows. For α < π/4, the minimal degree is clearly 2 but in general we are not able to prove exact lower bounds.
Whenever (1.1) holds, then the series expansion off representingϕ= 1 for Re z >0 andϕ= 0 for Rez <0 converges inp(K),and ifσ(A)⊂K then we do obtain a convergent expresion for the projection onto to unstable invariant subspace. However, wheneverp(A) would be nonnormal, it could happen that Vp(A)∩iR6=∅, and then we would not get a bound for the projection by the generalization of von Neumann theorem. To overcome this, note that from the
spectral radius formular(B) = limkBnk1/n in such a case, there does exist an integernsuch that with q =pn we have Vq(A)∩iR=∅.
This leads to the other topic discussed in this paper. With q = pn, the new variable there are no longer simple zeros and we shall therefore derive the multicentric representations needed in this case. This is done in Section 2 while the model problem is discussed in Section 3. In Section 4 we present concluding remarks and in the end we discuss a nonnormal small dimensional problem with q(z) = (z2−1)n.
2. REPRESENTATIONS AND MAIN ESTIMATES
2.1. Formulas and estimates. We need the basic formula for expressing a given function ϕ(z) as a linear combination of functions fj,k(wn) when w = p(z),forn a given positive integer.
Letp(z) be the monic polynomial of degreedwith distinct rootsλ1, . . . , λd. We denote by δk∈Pd−1 the Lagrange interpolation basis polynomials at λj
δk(λ) = p0(λ1k)
Y
j6=k
(λ−λj).
Then the multicentric representation of ϕtakes the form
(2.1) ϕ(z) =
d
X
j=1
δj(z)fj(w), where w=p(z) and fj’s are obtained from ϕwith the formula [7]
(2.2) fj(w) =
d
X
l=1
δl(λj, w)ϕ(ζl(w)), whereζl(w) denote the roots of p(λ)−w= 0 and
δl(λ, w) = p(λ)−w p0(ζl(w))(λ−ζl(w)).
Whenϕis holomorphic,f can also be computed from the Taylor coefficients of ϕat the local centersλj. In fact,
fj(w) =
∞
X
n=0 1
n!fj(n)(0)wn, wherefj(n)(0) can be computed recursively:
(p0(λj))nfj(n)(0) = (2.3)
=ϕ(n)(λj)−
d
X
k=1 n−1
X
m=0 n m
δ(n−m)k (λj)
m
X
l=0
bml(λj)fk(l)(0)−
n−1
X
l=0
bnl(λj)fj(l)(0).
Here the polynomials bnm are determined by bn+1,m =bn,m−1p0+b0nm
withbn0 = 0, b1,1 =p0 and bnm= 0 for m > n, see Proposition 4.3 in [9].
Remark. Erratum: the last term on the right of (2.3) is missing from the
formula (4.2) of [9].
Since the computations forfj’s are done with power series, we can move to p(z)n =wn,because the expansions are done for that variable. Therefore we formulate the next theorem.
Theorem2.1. Suppose p has simple zeros and assumeϕis holomorphic in a neighborhood of V(p, ρ) ={z∈C : |p(z)| ≤ρ} and given in the form
ϕ(z) =
d
X
j=1
δj(z)fj(w), where w=p(z).
Then
ϕ(z) =
d
X
j=1
δj(z)[fj,0(wn) +· · ·+wn−1fj,n−1(wn)],
where fj,k are holomorphic in a neighborhood of the disc|w| ≤ρ and given by
wkfj,k(wn) = n1nfj(w)+e−2πik/nfj(e2πi/nw)+· · ·+e−2πi(n−1)k/nfj(e2πi(n−1)/nw)o. In order to prove this, we consider a fixed functionfj and putg=fj.Then we set
wkgk(wn) = n1ng(w) +e−2πik/ng(e2πi/nw) +· · ·+e−2πi(n−1)k/n
g(e2πi(n−1)/nw)o
pointwise.
Proposition 2.2. Given an arbitrary n ∈ N and the functions gi, i = 0, . . . , n−1,for all w∈C we have:
g(w) =g0(wn) +wg1(wn) +· · ·+wn−1gn−1(wn).
Proof. Using the above formula ofgk(wn),fork= 0,1, . . . , n−1,we have
g0(wn) =n1ng(w) +g(e2πi/nw) +g(e4πi/nw) +· · ·+g(e2πi(n−1)/nw)o wg1(wn) =n1ng(w) +e−2πi/ng(e2πi/nw) +e−4πi/ng(e4πi/nw) +. . .
+e−2πi(n−1)/ng(e2πi(n−1)/nw)o
w2g2(wn) =n1ng(w) +e−4πi/ng(e2πi/nw) +e−8πi/ng(e4πi/nw) +. . . +e−4πi(n−1)/ng(e2πi(n−1)/nw)o
. . .
wn−1gn−1(wn) =n1ng(w) +e−2πi(n−1)/ng(e2πi/nw) +e−4πi(n−1)/ng(e4πi/nw) +· · ·+e−2πi(n−1)2/ng(e2πi(n−1)/nw)o.
Summing up all the terms we get
1
nng(w) +n1g(e2πi/nw)
n−1
X
k=0
e−2πik/n+ 1ng(e4πi/nw)
n−1
X
k=0
e−4πik/n+. . . +n1g(e2πi(n−2)/nw)
n−1
X
k=0
e−2πi(n−2)k/n+ 1ng(e2πi(n−1)/nw)
n−1
X
k=0
e−2πi(n−1)k/n
=g(w)
since all the other terms sum up to zero.
Proof of Theorem 2.1. It follows now immediately from Proposition 2.2 to-
gether with the following proposition.
Proposition 2.3. If fj is given for |p(z)| ≤ρ, then fj,k, k = 1, . . . , n−1, are defined for |p(z)n| ≤ρn and
(2.4) fj(p(z)) =
n−1
X
k=0
p(z)kfj,k(p(z)n), for |p(z)| ≤ρ.
Further, if fj(p(z)) is analytic for|p(z)| ≤ρ then so are fj,k(p(z)n).
Proof. First part is proved in Proposition 2.2.
For the second part, we assume fj analytic, thus it can be written as a power series
(2.5) fj(p(z)) =
∞
X
m=0
αmp(z)m.
We know that pointwise we have
p(z)kfj,k(p(z)n) = n1hfj(p(z)) +e−2πik/nfj(e2πi/np(z)) +. . . +e−2πi(n−1)k/n
fj(e2πi(n−1)/np(z))i. (2.6)
When we substitute (2.5) in (2.6), we get fj,0(p(z)n) =
= n1 hα0+α1p(z) +α2p(z)2+· · ·+αnp(z)n+. . .
+α0+e2πi/nα1p(z) +e4πi/nα2p(z)2+· · ·+αnp(z)n+. . . +. . .
+ α0+e2πi(n−1)/nα1p(z) +e4πi(n−1)/nα2p(z)2+· · ·+αnp(z)n+. . .i. Thusfj,0(p(z)n) =α0+αnp(z)n+α2np(z)2n+. . . , since all the other terms vanish.
We continue with p(z)fj,1(p(z)n),so we get
p(z)fj,1(p(z)n) =1nhα0+α1p(z) +α2p(z)2+· · ·+αnp(z)n+. . . +e−2πi/nα0+α1p(z) +e2πi/nα2p(z)2
+· · ·+e2πi(n−1)/nαnp(z)n+αn+1p(z)n+1+. . . +. . .
+e−2πi(n−1)/nα0+α1p(z) +e2πi(n−1)/nα2p(z)2 +· · ·+e2πi(n−1)2/nαnp(z)n+αn+1p(z)n+1+. . .i. Thereforefj,1(p(z)n) =α1p(z) +αn+1p(z)n+1+α2n+1p(z)2n+1+. . . . In a similar way it follows that
(2.7) p(z)kfj,k(p(z)n) =αkp(z)k+αn+kp(z)n+k+α2n+kp(z)2n+k+. . . Because all the coefficientsαmk,formk=k, n+k,2n+k,3n+k, . . . ,come from fj which is analytic, we know that lim sup|αmk|1/mk ≤ 1ρ, therefore we have thatp(z)kfj,k(p(z)n) is analytic, i.e. a converging power series.
Now, if we factor (2.7)
p(z)kfj,k(p(z)n) =p(z)khαk+αn+kp(z)n+α2n+kp(z)2n+. . .i we have that
fj,k(p(z)n) =αk+αn+kp(z)n+α2n+kp(z)2n+. . . is a converging power series, thus is analytic for |p(z)n| ≤ρn.
Next we need to be able to boundϕin terms offj,k’s and vice versa. The first one is straightforward.
Proposition 2.4. Denote L(ρ) = sup|p(z)|≤ρPdj=1|δj(z)|.Then sup
|p(z)|≤ρ
|ϕ(z)| ≤L(ρ) max
1≤j≤d n−1
X
k=0
sup
|w|≤ρ
|wkfj,k(wn)|.
The other direction is more involved and we formulate it in the following theorem.
Theorem 2.5. Assume p is a monic polynomial of degree d with distinct roots ands(ρ)denotes the distance from the lemniscate|p(z)|=ρto the nearest critical point of p, zc, such that |p(zc)|> ρ . Then there exists a constant C, depending on p but not on ρ, such that if ϕ is holomorphic inside and in a neighborhood of the lemniscate, then each fj,k is holomorphic for |w| ≤ρ and for |w| ≤ρ, 1≤j≤d,0≤k≤n−1, we have
(2.8) |wkfj,k(wn)| ≤ 1 + C
sd−1
sup
|p(z)|≤ρ
|ϕ(z)|.
For the proof of this statement we need some lemmas. The aim is to bound the representing functionsfj,k in terms of the original functionϕ. To that end we first quote the basic result of bounding fj in terms ofϕ and then proceed boundingfj,k in terms offj.
Lemma 2.6. (Theorem1.1 in [7]) Suppose ϕ is holomorphic in a neighbor- hood of the set {ζ :|p(ζ)| ≤ρ} and let sbe as in Theorem 2.5. There exists a constant C, depending on p but independent ofρ andϕ, such that
sup
|w|≤ρ
|fj(w)| ≤ 1 + C
sd−1
sup
|p(z)|≤ρ
|ϕ(z)|, for all j = 1, . . . , d.
Lemma 2.7. In the notation above we have sup
|w|≤ρ
|wkgk(wn)| ≤ sup
|w|≤ρ
|g(w)|.
Proof. We have
wkgk(wn) = n1{g(w)+e−2πik/ng(e2πi/nw)+· · ·+e−2πi(n−1)k/n
g(e2πi(n−1)/nw)}.
The bound follows by taking the absolute values termwise.
Proof of Theorem 2.5. The proof follows immediately from these two lemmas.
Remark 2.8. Gauss-Lucas theorem asserts that given a polynomialpwith complex coefficients, all zeros ofp0 belong to the convex hull of the set of zeros ofp,see [3]. Thus, as soon asρis large enough, all the critical points will stay inside the lemniscate whenever the lemniscate is just a single Jordan curve.
But when we want to make the separation, we start squeezing the level of the lemniscate and this results in leaving at least one critical point outside.
Therefore the s we are measuring is the distance to the boundary from that particular critical point.
If the critical points ofpare simple (as they generically are) then the depen- dency of the distance is inverse proportional and the coefficient in the theorem takes the form
1 +Cs.
To see how one can find these constants C and what they describe we will present an example with the computations ofδl(λj, w) forp(z) =z4+ 1.These computations will be used in computing the constants. We also apply them for the Riesz projections.
Example 2.9. Let p(z) = z4 + 1 with roots λ1 = (−1)1/4, λ2 = (−1)3/4, λ3 = −(−1)1/4 and λ4 = −(−1)3/4. Let w = z4+ 1. Using the formula for δl(λj, w) we have:
δ1(λ1, w) = 1 +38w+1964w2+12833w3+. . . δ1(λ2, w) = −1+i8 w− 3+4i32 w2−20+31i256 w3+. . . δ1(λ3, w) = −18w−647w2−12813w3+. . . δ1(λ4, w) = −1−i8 w− 3−4i32 w2−20−31i256 w3+. . . δ2(λ1, w) = −18w−647w2−12813w3+. . . δ2(λ2, w) = −1−i8 w− 3−4i32 w2−20−31i256 w3+. . . δ2(λ3, w) = 1 +38w+1964w2+12833w3+. . . δ2(λ4, w) = −1+i8 w− 3+4i32 w2−20+31i256 w3+. . . δ3(λ1, w) = −1−i8 w− 3−4i32 w2−20−31i256 w3+. . . δ3(λ2, w) = 1 +38w+1964w2+12833w3+. . . δ3(λ3, w) = −1+i8 w− 3+4i32 w2−20+31i256 w3+. . . δ3(λ4, w) = −18w−647w2−12813w3+. . . δ4(λ1, w) = −1+i8 w− 3+4i32 w2−20+31i256 w3+. . . δ4(λ2, w) = −18w−647w2−12813w3+. . . δ4(λ3, w) = −1−i8 w− 3−4i32 w2−20−31i256 w3+. . . δ4(λ4, w) = 1 +38w+1964w2+12833w3+. . .
whereζl(w),forl= 1,4,are given byζ1(w) = (w−1)1/4, ζ2(w) =−(w−1)1/4, ζ3(w) =i(w−1)1/4 and ζ4(w) =−i(w−1)1/4.
Remark 2.10. From Lemma 2.6 we see that (2.9) |δ(m)l (λk, w)|∼ Cm
sm+1.
where m is the multiplicity of the nearest critical point of p outside the lem-
niscate.
To be able to compute the constantsC from equation (2.9) for polynomials of degree d ≥ 4, we need to separate the spectrum by perturbing the roots with an angleεsmall enough and by dropping the magnitude ρbelow 1.The perturbations for polynomials of degree 4,6,8,10,12 and 14 are described in the next section.
Note that before the perturbation we have multiple critical points, all inside the lemniscate, except for 0,which is on the level curve. After we perturb the roots, one critical point is left outside the lemniscate, while all the others remain inside. Therefore, in our case, the multiplicity mis zero.
From now on we will choose a random value forεto make some experimental computations. All the computations below will work properly for any other random valueεsmall enough that ensures the desired separation.
Now, if we choose a random perturbation with, for example, ε=π/70,we get the following values for the constant C
Degree 4 6 8 10 12 14
Constant C 576.4344 1.4665 8.0721 2.2754 12.8520 4.0475 Table 2.1. The constantC.
The computations for P|δl(λk, w)| were made with Mathematica and the values for s, the smallest distance from the lemniscate to the nearest critical point, were computed with the help of Tiina Vesanen that provided a Matlab program. The codes can be found in the appendix of [1], which is a preprint version on this article, that contains in appendix material which is not included in this paper. For all these computations one has to choose a value for the levelρ, smaller than 1. If one chooses levelρ= 1,then the value forswill be zero, since the lemniscate passes through origin. Therefore we have chosen the minimum value for the levelρ such that the lemniscate separates only in two parts when having a perturbation with ε = π/70. Hence we have registered the following data
Degree 4 6 8 10 12 14
P|δl(λk, w)| 2293.81 6.5122 46.6599 16.3586 83.4547 16.7046 s 0.2513 0.2252 0.1730 0.1391 0.1540 0.2423 Table 2.2. Experimental values which help in computing the constant C in Table 2.1.
From the table of constants C one can see that the lemniscate bifurcates differently even with a small ε.
Remark 2.11. For the quadratic polynomial p(z) = z2 −1, one does not need to perturb the roots, but just to decrease the magnitude ofρbelow 1.In
this case, for example, if the level is ρ = 0.9 then one gets C = 0.2, while if
the level is ρ= 0.99 thenC= 0.6956.
2.2. Application to Riesz projection. Let A be a bounded operator in a Hilbert space such that
Vpn(A) ={z : |p(z)n| ≤ kp(A)nk}.
In order to compute the Riesz projection we takeϕ= 1 in one of the compo- nents andϕ= 0 in the other components of Vpn(A) and
fj(p(z)) =
n−1
X
k=0
p(z)kfj,k(p(z)n).
We shall apply the following theorem, if ϕ is holomorphic in a neighbour- hood of the unit disk Dand A∈ B(H),then
(2.10) kϕ(A)k ≤sup
D
|ϕ|, see e.g. [11].
From Theorem 2.5 we have for|p(z)| ≤ρthat (2.11) |p(z)kfj,k(p(z)n)| ≤1 +sd−1C
sup
|p(z)|≤ρ
|ϕ(z)|,
and since fj,k are analytic, we can apply (2.10) to each of them, so with ρ=kp(A)nk1/n
(2.12) kfj,k(p(A)n)k ≤ sup
|p(z)|≤ρ
|fj,k(p(z)n)|.
We have,
kp(A)kfj,k(p(A)n)k ≤ kp(A)kkkfj,k(p(A)n)k and from (2.12) we get
kp(A)kfj,k(p(A)n)k ≤ kp(A)kk sup
|p(z)|≤ρ
|fj,k(p(z)n)|.
By the Maximum principle we see that kp(A)kk sup
|p(z)|≤ρ
|fj,k(p(z)n)|=kp(A)kkρ−k sup
|p(z)|≤ρ
|p(z)kfj,k(p(z)n)|.
Therefore, by (2.11), we have
kp(A)kfj,k(p(A)n)k ≤ k(p(A)/ρ)kk1 +sd−1C sup
|p(z)|≤ρ
|ϕ(z)|.
Substituting now in the decomposition (2.1), ϕ(A) becomes the Riesz pro- jection and it is bounded by
(2.13) kϕ(A)k ≤
1 +sd−1C kp(A)ρkkk
d
X
j=1
kδj(A)k
sup
|p(z)|≤ρ
|ϕ(z)|.
Thus we have proven the following theorem.
Theorem2.12. Given a polynomialpof degreed, with distinct roots and a bounded operator A in a Hilbert space, we assume that the "expression"
Vpn(A) ={z : |p(z)n| ≤ kp(A)nk}
has at least two components. Setρ=kp(A)nk1/n and letϕ be a function such that ϕ= 1in one component of Vpn(A) andϕ= 0in the others. Let sbe the distance from the nearest outside critical point to the boundary of Vpn(A).
Then, considering fj,k as given by Theorem 2.1, one has ϕ(A) =
d
X
j=1
δj(A)
n−1
X
k=0
p(A)kfj,k(p(A)n),
which is the Riesz spectral projection onto the invariant subspace corresponding to the spectrum inside the component where ϕ= 1.
The bound for the norm of ϕ(A) is given by(2.13).
Remark 2.13. In applications we will consider ϕ = 1 in the components of Vpn(A) that are on the right complex half-plane and ϕ =−1 in the com- ponents on the left complex half-plane. This way the computations are more symmetrical. In this situation there is only one critical point at the origin, which is simple, so we will have 1 +Cs in the formula for the bound of the
Riesz projection.
3. SEPARATING POLYNOMIALS
3.1. Separation tasks. In this section we shall discuss separating issues by lemniscates. To that end, given a polynomial p denote byV(p, ρ) the set
V(p, ρ) ={z∈C :|p(z)| ≤ρ}.
A key result in this context is the following. LetK⊂Cbe compact and such that C\K is connected. For δ >0 denote further
K(δ) ={z : dist(z, K)< δ}.
Then there exists a polynomial p and ρ >0 such that K ⊂V(p, ρ)⊂K(δ).
In particular, if K =K1∪K2 and K1∩K2 =∅ sinceK is compact, then for small enoughδ
K1(δ)∩K2(δ) =∅
as well. Thus, V(p, ρ) separates the components K1 and K2 respectively.
Suppose that we have two analytic functions ϕj, each analytic in Kj(δ). We can view them as just one analytic function
ϕ:K(δ)→C,
where ϕ agrees to ϕj on Kj(δ). We are interested in particular in the case where ϕj is constant. Multicentric representation then gives a power series which is simultaneously valid in both components.
So, we can ask, for such a separation task what is the smallest degree of a polynomial achieving this.
We model this as follows: Let
K1(δ) ={z= 1 +iy :|y| ≤tan(δ)}
and K2(δ) symmetrically on the other side of the imaginary axis:
K2(δ) ={z=−1 +iy :|y| ≤tan(δ)}
Our first problem concerns the minimal degree of a polynomialpsuch that Kj(δ)⊂V(p, ρ)
and
V(p, ρ)∩iR=∅.
This is discussed in the next subsection.
Another natural task is related to existence of a logarithm. So,C is again compact and we assume 0 ∈/ C. Now there exists a single valued logarithm in C if and only if 0 is in the unbounded component of the complement of C. That is, the setC does not separate origin from infinity. The natural task here is to find a polynomial p such that
C ⊂V(p, ρ) and
0∈/V(p, ρ).
AsV(p, ρ) is polynomially convex, this suffices for representing the logarithm inV(p, ρ).
3.2. Model problem. Let p(z) = zd−1 be a monic polynomial of complex variablez, with|p(z)|= 1.The polynomial pcan be written as
p(z) =
d
Y
j=1
(z−eiθj)
where eiθj are the roots of p and θj are the angles of the roots. For d= 4m, m≥1 we have
p(z) =
m
Y
j=1
(z−eiθj)
m
Y
j=1
(z−e−iθj)
m
Y
j=1
(z+eiθj)
m
Y
j=1
(z+e−iθj).
We are interested to separate the spectrum ofp,as discussed earlier in this paper. In this sense, we first check if there are roots laying on the imaginary axis. If so, a rotation with angleπ/dis applied so that no roots touch iR.
Next we perturb the roots as follows: the four roots that are closest to the imaginary axis (complex roots) are moved along the unit circle towards the real axis with small angleε.Then the level ρis slightly decreased withη.This approach is used for polynomials of degree d ≥4. The quadratic polynomial
case is shortly discussed below and cubic polynomial case is presented as a first example.
To this end we have to find the maximum η for a chosen ε so that the spectrum gets separated in only two parts, one on the right hand side of the imaginary axis and one on the left hand side. Also we can find the maximum angleα (see Figure 3.1) such that the spectrum will lay inside our lemniscate.
Fig. 3.1. Level 0.996.
Now we analyze some cases. The quadratic polynomial,p(z) =z2−1 is the classical lemniscate and for this one just has to decrease the magnitude ρ to below 1. No perturbation of the roots is needed. For a decrease withη= 0.01 of the level we get the picture from Figure 3.2:
−2 −1 0 1 2
−2
−1.5
−1
−0.5 0 0.5 1 1.5 2
y
x
Fig. 3.2. Level 0.99.
Example 3.1. Letp(z) =z3−1 with rootse2πi/3, e−2πi/3 and 1.We apply a perturbation with εto the complex roots and we get
pε(z) = (z−ei(2π/3+ε))(z−e−i(2π/3+ε))(z−1).
Thus the lemniscate is |pε(z)|= 1−η. For ε= π/70 the resulted picture is
shown in Figure 3.3.
−1.5 −1 −0.5 0 0.5 1 1.5
−1
−0.5 0 0.5 1
y
x
(a) Level 1.
−1.5 −1 −0.5 0 0.5 1 1.5
−1
−0.5 0 0.5 1
y
x
(b) Level 0.996.
Fig. 3.3. Separation of the cubic polynomial.
Example 3.2. Let p(z) = z4−1. Since there are roots on the imaginary axis, we have to apply a rotation with π/4. Thus our polynomial becomes p(z) =z4+ 1.
Then
p(z) = (z−eiθ)(z−e−iθ)(z+eiθ)(z+e−iθ) whereθ=π/4,so we have
(3.1) p(z) = (z2−e2iθ)(z2−e−2iθ) =z4−2 cos(2θ)z2+ 1.
Next, we decrease the angle θ withεsmall enough and denote the new angle θε=θ−ε. For the polynomial with perturbed θwe compute the lemniscate.
We have that
e2iθε = e2iθ−2iε=eiπ/2−2iε
= i(1−2iε−2ε2+. . .) = 2ε+i(1−2ε2+. . .).
Forz=t∈R
|pθε(t)| = |t2−2ε+i(−1 + 2ε2+. . .)|2
= t4−4t2ε+ 1 +o(ε2).
(3.2)
Forz=it∈C
|pθε(it)| = |(it)2−2ε+i(−1 + 2ε2+. . .)|2
= t4−4t2ε+ 1 +o(ε2).
We writet=z=x+iyin (3.2) and compute the lemniscatel:|pε(z)|= 1−η, wherepε(z) =z4−2 cos(2θε)z2+ 1 from (3.1). Thus
l: (x2+y2)4+ 2(x2+y2)2−16x2y2+ 16ε2(x2+y2)2
−8ε(x6+x4y2−x2y4−y6+x2−y2) = 0.
Forε=π/70 the results are shown in Figure 3.4.
−1 −0.5 0 0.5 1
−1
−0.5 0 0.5 1
y
x
original root perturbed root critical point
(a) Level 1.
−1 −0.5 0 0.5 1
−1
−0.5 0 0.5 1
y
x
original root perturbed root critical point
(b) Level 0.997.
Fig. 3.4. Separation of the quartic polynomial.
Similar computations were made for polynomials of degree 6,8,10,12 and 14,and these can be seen in [1].
Remark 3.3. The goal was to find the maximum angle α such that the spectrum lays inside the lemniscate. For this, one can compute the ratioa/b, where aand b are the length of the linesa and b from Figure 3.5 below, and hence the angleα is
α= arctan(a/b).
Note that, for a minimum level one might have that the line a cuts the lemniscate, situation that happens even for a slightly perturbation of the level in all the cases with polynomials of degreed≥6.Therefore, one has to consider a smaller angle.
Fig. 3.5. Ratio and angle α.
For a minimum level ρ of the lemniscate and a perturbation withε=π/70 we have the values for the ratio from Table 3.1 or from the chart in Fig. 3.6.
Degree 4 Degree 6 Degree 8 Degree 10 Degree 12 Degree 14
a 0.5637 0.9040 0.9905 1.0043 0.9973 0.9846
b 0.3090 0.3767 0.3790 0.3624 0.3402 0.3176
a/b 1.8242 2.3997 2.6134 2.7712 2.9315 3.1001 Table 3.1. Ratio.
Fig. 3.6. Ratio chart.
The maximum angle α that we have found is presented in the chart from
Figure 3.7.
Fig. 3.7. Maximum angleα.
Remark 3.4. The quadratic polynomial is a special case since the only perturbation applied is decreasing the level. In this case, the maximum angle is when the levelρ is unchanged, in this caseα= 45◦ and the minimum angle goes to 0◦when the level is significantly decreased withη= 0,99.For a slightly change withη= 0.01 we have found an angle of 29.92◦ and for a change with η= 0.1 we have registered an angle of 26.57◦. Remark 3.5. For polynomials of degree d ≥ 4, it is easy to check the maximum value of η, i.e. the minimum value that the level can have, such that we get the desired separation. For example, if again the perturbation is ε=π/70 we get the values from Table 3.2.
Degree 4 6 8 10 12 14
η 0.008 0.0078 0.0038 0.0021 0.0022 0.0045 Table 3.2. Maximum changeη of the levelρ.
With these values the lemniscate squeezes next to the closest critical point to the perturbed root. A bigger decrease of the level would force that critical point to get out from the interior of the lemniscate and thus one doesn’t get the desired separation. These estimates may not be the best but they are what we have reached by manipulating the pictures and the resulting pictures
can be seen in [1].
4. CONCLUDING REMARKS
In Section 2 we presented the closed formula and the bound for the Riesz projection, in Section 3 we described the separation process and in this section one can see how the expansions on the Riesz projection look like when taking ϕ= 1 for the components on the right side of the imaginary axis andϕ=−1 for the ones on the left side. We will finish this paper with an example that shows what the effects are on the Riesz projection.
In the appendices of [1] one can find applications that explicitly compute the series expansions for fj’s in the decomposition
ϕ(z) =
d
X
j=1
δj(z)fj(w), w=p(z),
when ϕ is identically 1 on the right half plane and−1 on the left half plane, for the quadratic, the quartic, the perturbed quartic polynomial and the poly- nomial q(z) =p(z)n= (z2−1)n, respectively.
Remark 4.1. In using multicentric calculus a central problem is to find a polynomial p such that p(A) has small norm and, when aiming for Riesz projection, that the lemniscate on the level ofkp(A)kseparates the spectrum into different components. This can be done, for example, by minimizing kp(A)k approximately over a set of polynomials, or, by using a suitable p which has been computed for a neighbouring matrix.
Alternatively, and that is the main topic here, one search for polynomials p such that it is small in a neighbourhood of the spectrum of A. And then computes high enough powerp(A)2m such thatkp(A)2mk1/2m ∼ρ(p(A)).
In the following example we point out with the help of a low-dimensional problem, how the size of coupling can affect on the need of taking a high power of p(A).
Example 4.2. Let
(4.1) A=
B X 0 −B
be a 4×4 matrix where
(4.2) B =
α 1 0 α
and
(4.3) X=
0 γ γ 0
.
In this example we could take p(z) =z2−α2 to actually get a closed form for the projection. However, we take p(z) =z2−1 as our polynomial and then the effect ofα >0 being close or further away from 1 models the lack of exact knowledge on the spectrum. We are interested in havingkp(A)nk<1 and ask how the parametersαandγ contribute to the value ofnneeded. Qualitatively it is clear that such annexists if and only if α <√
2, independently of γ.
Substituting Ainto pwe have p(A) =
C Y
0 C
,
where
C=
α2−1 2α 0 α2−1
and
Y =
γ 0 0 −γ
. A short computation shows that
p(A)n=
Cn n(α2−1)n−1Y
0 Cn
. Thus, we have
kp(A)nk ∼ |α2−1|n−1h|α2−1|+n(|α|+|γ|)i,
so that if|α2−1| 1 then a small nshall work. If however,|α2−1|= 1−ε with 0 < ε 1, modelling the case when e.g. spectrum of A is scattered inside the lemniscate, then the behavior is of the nature
kp(A)nk ∼(1−ε)n(n+ 1),
which becomes below 1 only for n1/ε.
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Received by the editors: November 9, 2015.
