Research ArticleComputing the q -Numerical Range of Differential Operators

Ahmed Muhammad 1 and Faiza Abdullah Shareef1

1Department of Mathematics, College of Science, University of Salahaddin, Erbil, Iraq

Correspondence should be addressed to Ahmed Muhammad; ahmed.muhammad@su.edu.krd

Received 7 May 2020; Accepted 17 July 2020; Published 28 August 2020

Academic Editor: Ali R. Ashrafi

Copyright © 2020 Ahmed Muhammad and Faiza Abdullah Shareef. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided theoriginal work is properly cited.

A linear operator on a Hilbert space may be approximated with finite matrices by choosing an orthonormal basis of thez Hilbertspace. In this paper, we establish an approximation of the q-numerical range of bounded and unbounnded operator matrices byvariational methods. Application to Schrödinger operator, Stokes operator, and Hain-Lüst operator is given.

1. Introduction and Definitions

The simplest concepts which can be used to obtain an enclo-sure of the spectrum of a linear operator A in a Hilbert spaceH are the numerical range.

W Að Þ = Ax, xh i,  for some x ∈D Að Þ,∥x∥ = 1 =f g, ð1Þ

where Dð:Þ denotes the domain. It is not difficult to see thatthe point spectrum σpðAÞ of A is contained inWðAÞ and thatthe approximate point spectrum σappðAÞ of A is contained in

the closure ofWðAÞ; the inclusion σðAÞ ⊂ �WðAÞ holds if A isclosed. Many estimates of eigenvalues of differential opera-tors, for instance, involve calculating estimates of the innerproducts hAx, xi, using partial integration. It is always con-vex [1]. However, the numerical range often gives a poorlocalization of the spectrum and cannot reveal the existenceof spectral gaps. In [2], the numerical range of a (finite)matrix was approximated by projection methods. This con-cept was generalized to q -numerical range in [3], as follows:

Wq Að Þ = Ax, yh i,  for some x, y ∈H,∥x∥ = 1 = ∥y∥, fx, yh i = qg, ð2Þ

where 0 ≤ q ≤ 1. It is easy to see that if q = 1, then WqðAÞcoincides with the numerical range of A. For a closed opera-tor A, the closure of the q -numerical range contains the

eigenvalues of A scaled by q, and WqðαA + βIÞ = αWqðAÞ +βI, for every α, β ∈ℂ. Moreover, It is known that WqðAÞ isa compact convex subset of ℂ [4]. A review of the propertiesof the q -numerical range of operator matrices may be foundin [3, 5, 6]. The main purpose of this work is the approxima-tion of the q -numerical range of bounded and unboundedoperator matrices, which contains both theoretical resultsand applications to some self-adjoint and non-self-adjointoperator matrices. The paper is organized as follows. In Sec-tion 2, we establish an approximation of the q -numericalrange of bounded and unbounded operator matrices. In Sec-tion 3, we shall apply these results to compute the q -numer-ical range of differential operators and some new analyticbounds.

2. Convergence Theorems

In this section, we will use finite matrices to approximate thenumerical range of linear operators. However, the idea ofapproximating linear operators by finite matrices is an obvi-ous one that must happen again and again, suppose that onewishes to compute the q -numerical range of A by using thefollowing projection method. Let ðLkÞ∞k=1 be a nested familyof spaces in H given by Lk = spanfϕ1, ϕ2,⋯, ϕkg, where fLk : k ∈ℕg is the orthonormal basis of a Hilbert space H

whose element lies in DðAÞ and suppose that the corre-sponding orthogonal projections Pk : H ⟶Lk convergestrongly to the identity operator I. We identify A with its

HindawiJournal of Applied MathematicsVolume 2020, Article ID 6584805, 12 pageshttps://doi.org/10.1155/2020/6584805

matrix representation with respect to fϕk : k ∈ℕg:

A = Ai j

� �∞i,j=1, Ai j ≔ Aϕ j, ϕi

D E i, j ∈N: ð3Þ

Then, the compression of A to Lk is denoted byAk ≔ PkAjLk

, where

Ak =

Aϕ1, ϕ1h i Aϕ1, ϕ2h i ⋯ Aϕ1, ϕkh iAϕ2, ϕ1h i Aϕ2, ϕ2h i ⋯ Aϕ2, ϕkh i⋮ ⋮ ⋮

Aϕk, ϕ1h i Aϕl, ϕ2h i ⋯ Aϕk, ϕkh i

0BBBBBBBB@

1CCCCCCCCA: ð4Þ

Theorem 1. Let A be a bounded operator in H Let fLk : k∈ℕg be a nested family of spaces in H given by Lk = spanfϕ1, ϕ2,⋯, ϕkg where fϕk : k ∈ℕg is orthonormal, and Akbe as in Equation((4)).Then, WqðAkÞ ⊆WqðAÞ ,for q ∈ ½0, 1�:

Proof. Define an isometry i : Lk ⟶ℂk by iðα1ϕ1 + α2ϕ2+⋯+αkϕkÞ≔ ðα1, α2,⋯,αkÞ, and iðβ1ϕ1 + β2ϕ2+⋯+βkϕkÞ≔ ðβ1, β2,⋯,βkÞ: Suppose that λ ∈WqðAkÞ: Then, for some α,β ∈ℂk, with ∥α∥ = 1 = ∥β∥,ðα, βÞ = q such that λ = hAkα, βi:Choose ξ, η ∈Lk, such that iðξÞ = α,iðηÞ = β, and ∥ξ∥ = 1 = ∥η∥: Then, a direct computation shows that λ = hAξ, ηi, whereξ =∑k

j=1 αjϕj, and η =∑kj=1 βjϕj: Thus, λ ∈WqðAÞ:

The next inclusion, which will be used in the proof ofTheorem 3, asserts that fðWqðAkÞÞ: k ≥ 2g forms an increas-ing sequence of sets.

Lemma 2. Let fLk : k ∈ℕg andAk be as in Theorem1. Givenq ∈ ð0, 1�, then WqðAkÞ ⊆WqðAτÞ, for r ≥ k.

Proof. This is an immediate consequence of the fact thatℂk isa subspace of ℂτ. In detail, suppose λ is in WqðAkÞ; then,there exists α, β ∈ℂk, ∥α∥ = 1 = ∥β∥, with hα, βi = q such thatλ = hAkα, βi: Choose ς, ν ∈ℂτ by setting ς =ðα1, α2,⋯,αj, 0,⋯,0ÞT , ν = ðβ1, β2,⋯,βj, 0,⋯,0ÞT . A simplecalculation shows that hς, νi = hα, βi = q and hAkα, βi = hAkς, νi and so λ is in WqðAτÞ.

Theorem 3. Let A and Ak be as in Theorem1. Let Pk : H→Lk be orthogonal projection. If Pk converge strongly tothe identity operator I, then �WqðAÞ = �S∞

j=1 WqðAjÞ.

Proof. In view of Theorem 1, it is sufficient to prove WqðAÞ⊆ �S∞

j=1 WqðA jÞ. Suppose λ ∈WqðAÞ. Choose x, y ∈H such

that ∥x∥ = ∥y∥ = 1 and hx, yi = q, such that λ = hAx, yi. Weknow Pkx⟶ x and Pky⟶ y as k⟶∞ and if xk ≔ iðPkxÞ,yk ≔ iðPkyÞ thus hAxk, yki⟶ hAx, yi as k⟶∞ and hxk,yki⟶ hx, yi = q as k⟶∞.

Fix k > 0: Let i : spanfϕ1, ϕ2,⋯, ϕskg⟶ℂsk be the stan-

dard isometries as in the proof of Theorem 1. Define ~αk, ~βk

∈ Csk by ~αk = iskðxkÞ,~βk = iskðykÞ: Consider the sk × sk matrixAsk

, where the ðp, rÞ-element of the sk × sk matrixAskis equal

to hAϕp, ϕri, for p, r = 1, 2,⋯, sk. A simple calculation shows

that hAsk~αk, ~βki = hAxk, yki, h~αsk , ~βsk

i = hxk, yki: Since hAxk,yki⟶ hAx, yi as k⟶∞ and hxk, yki⟶ hx, yi = q as k⟶∞, we have hAsk

~αk, ~βki⟶ hAx, yi as k⟶∞. Hence,there exist λk ∈WqðAsk

Þ such that λk ⟶ λ. In view of

Lemma 2, this immediately gives λ ∈ �S∞j=1 WqðAjÞ.

Remark 4. The hypotheses Pk converge strongly to the identityoperator I are (in general) necessary. It is easy to construct anexample where λ ∈ �S∞

j=1 WqðA jÞ is a strict subset of WqðAÞ.

Example 1. Let A be an operator matrix in H = ℓ2ðℕÞ, whereA = diag f1/ng∞n=1fLk : k ∈ℕg be a nested family of sub-space in ℓ2ðℕÞ, with Lk = span fe2,⋯, ek+1g, where ej = jth

standard basis vector, and rk = span fψ1,⋯, ψkg whereðψkÞ∞k=1 is any orthonormal sequence. Then, performing ananalysis analogous to Theorem3, we see that ðAxk, ykÞ is notconvergent to ðAx, yÞ, unless x is orthogonal to e1:

Remark 5. We assume readers are familiar with basic notionsand results about linear unbounded operators, as well asmatrices of nonnecessarily bounded operators. Useful refer-ences are [7–9]. We call a few definitions though: a linearoperator A with a domain DðAÞ contained in a Hilbert spaceH is said to be densely defined ifDðAÞ =H . Say that a linearoperatorA is closed if its graph ΓA is closed inH ⊕H .A linearoperator A is called closable if the closure �ΓA of its graph is thegraph of some operators. A subspaceD ⊂DðAÞ is called a coreof a closable operator A if AjD is closable with closure A. Thedefinition of the q -numerical range for bounded linear opera-tors in Equation((2)) generalizes as follows to unboundedoperator matrices A with dense domain DðAÞ.

Definition 6. For a linear operator A with domainDðAÞ ⊂H ,we define the q -numerical range of A for 0 ≤ q ≤ 1 by

Wq Að Þ = Ax, yh i,  for some x, y ∈D Að Þ,∥x∥ = 1 = ∥y∥, fx, yh i = qg: ð5Þ

Theorem 7. Let A be an unbounded operator in H . Let fLk : k ∈ℕg be a nested family of spaces in DðAÞ given byLk = spanfϕ1, ϕ2,⋯, ϕkg, where fϕk : k ∈ℕg is ortho-normal, and Ak be as in Equation((4)). Then, WqðAkÞ ⊆Wq

ðAÞ, for q ∈ ð0, 1�:

Proof. Define an isometry π : Lk →ℂk by πðζ1ϕ1 + ζ2ϕ2+⋯+ζkϕkÞ≔ ðζ1, ζ2,⋯,ζkÞ, and πðγ1ϕ1 + γ2ϕ2+⋯+γkϕkÞ≔ ðγ1, γ2,⋯,γkÞ: Suppose that λ ∈WqðAkÞ: Then, for some ζ, γ∈ℂk, with ∥ζ∥ = 1 = ∥γ∥,ðζ, γÞ = q such that λ = hAkζ, γi:Choose ξ, η ∈Lk, such that πðξÞ = ζ,πðηÞ = γ, and ∥ξ∥ = 1 =

2 Journal of Applied Mathematics

∥η∥: Then, a direct computation shows that λ = hAξ, ηi,where ξ =∑k

j=1 ζjϕj, and η =∑kj=1 γjϕj: Thus, λ ∈WqðAÞ:

The following lemma can be the proof in a similar fashionas Lemma 2.

Lemma 8. Let fLk : k ∈ℕg andAk be as in Theorem7. Givenq ∈ ð0, 1�, then WqðAkÞ ⊆WqðAτÞ, for r ≥ k.

In the following result, we describe that the closure of therangeWqðAÞ is approximated byWqðAkÞ under the assump-tion that the linear span of fϕ1, ϕ2,⋯g is a core of A.

Theorem 9. Let A be an unbounded operator in H . Let fLk : k ∈ℕg be a nested family of spaces in DðAÞ given byLk = spanfϕ1, ϕ2,⋯, ϕkg, where fϕk : k ∈ℕg is orthonor-mal, and Ak be as in Equation((4)). Then, �WqðAÞ =

�S∞k=1 WqðAkÞ, for q ∈ ½0, 1�:

Proof. Since C = spanfϕ1, ϕ2,⋯g is a core of A, there exists asequence ðxkÞ∞k=1, with each xk ∈ spanfϕ1, ϕ2,⋯, ϕskg forsome sk > 0 such that ∥x − xk∥⟶0 and ∥Ax − Axk∥⟶0. Ina similar way, we may also find a sequence ðykÞ∞k=1, with eachyk ∈ spanfϕ1, ϕ2,⋯, ϕskg for some sk > 0, such that ∥y − yk∥⟶0 and ∥Ay − Ayk∥⟶0, so this means that ∥hAxk, yki − hAx, yi∥⟶0 as k⟶∞ and ∥hxk, yki − q∥⟶0 as k⟶∞.Fix k > 0: Let π : spanfϕ1, ϕ2,⋯, ϕskg⟶ℂsk be the stan-

dard isometries as in the proof of Theorem 7. Define ~αk, ~βk

∈ Csk by ~αk = πskðxkÞ,~βk = πsk

ðykÞ: Consider the sk × sk matrixAsk

, where the ðp, rÞ-element of the sk × sk matrix Askis equal

to hAϕp, ϕri, for p, r = 1, 2,⋯, sk. A simple calculation shows

that hAsk~αk, ~βki = hAxk, yki and h~αk, ~βki = hxk, yki. Since ∥hA

xk, yki − hAx, yi∥⟶0 as k⟶∞ and ∥hxk, yki − q∥⟶0 ask⟶∞, this implies that ∥hAsk

~αk, ~βki − hAx, yi∥⟶0 as k⟶∞. Hence, there exists λk ∈WqðAsk

Þ such that λk ⟶

λ. In view of Lemma 8, this immediately gives λ ∈�S∞

k=1 WqðAkÞ.

3. Numerical Experiments onDifferential Operator

In this section, we study some concrete examples and dem-onstrate that, in spite of the results obtained in the previoussection, practical computation of the q -numerical range ofdifferential operator is very far from being straightforward.We define the inner product hu, vi to be linear in the firstparameter and conjugate linear in the second parameter,and we consider the space of square-integrable functions,L2ðΩ, dxÞ, where Ω is an interval in ℝ, a Hilbert space withinner product

u, vh i =ðΩ

u�vdx: ð6Þ

The computations were performed in Matlab.

3.1. Application to Schrödinger Operator. In the Hilbert spaceH ≔ L2ð0, 1Þ, we introduce the Schrödinger operator

A = −d2

dx2+ p ð7Þ

(with bounded potential p) and the domain of L is given by

D Að Þ = u ∈H2 0, 1ð Þ: u 0ð Þ = 0 = u 1ð Þ� �: ð8Þ

Remark 10.

(i) Because A is self-adjoint and bounded below withpurely discrete spectrum, the eigenvalues of A aregiven by

λk ≔ infF⊂D Að Þdim F=k

supy∈Fy≠0

g yð Þ,ð9Þ

where g is the Rayleigh functional

g yð Þ≔ Ay, yh iy, yh i , y ∈D Að Þ, y ≠ 0 ð10Þ

(ii) It is clear the operator A in L2ð0, 1Þ has a sequence ofeigenvalues and normalized eigenfunctions for theoperator A in L2ð0, 1Þ are

λn = n2π2, ð11Þ

ϕn xð Þ =ffiffiffi2

psin nπxð Þ for n = 1, 2, 3,⋯ ð12Þ

under the setting pðxÞ = 0

(iii) Because Equation (7) is a closed operator, then itis not difficult to see that the subspace CA =spanfϕ1, ϕ2,⋯g ⊂DðAÞ is a core of A, so in thiscase, the main Theorem 9 is applicable to thisexample

(iv) We may use these eigenfunctions in Equation(12) as basis elements for a discretization of thetype discussed in Section 2, form the matrix ele-ments h−ϕk ′′, ϕji, using the inner product inEquation (6) with respect to the orthonormalbasis in Equation (12) and consider the (infinite)operator matrix

3Journal of Applied Mathematics

Q≔ −ϕk ′′, ϕj

D Eð13Þ

The matrix Ak in Equation (4) is obtained by taking theleading submatrices of the (infinite) operator matrix in Equa-tion (13), with appropriate dimensions. Observe that h−ϕk ′′, ϕji = diag fπ2, 4π2, 9π2,⋯g which can be evaluatedexplicitly. The following figure shows attempts to calculateWqðAkÞ for various k and q and also some attempts to esti-mate these sets by qualitative means, using existing theoremsfrom the literature as well as the theorems proved above.

3.2. Analytical Estimates for Schrödinger Operator. In orderto understand to what extent Figure 1 is qualitatively correct,we now analyze the q-numerical range of the Schrödingeroperator. Comments:

(i) In Remark 10 part (i), it is obvious hAy, yi ≥ π2hy, yi;thus, the numerical range of A is �WðAÞ = ½π2,∞Þ

(ii) While for 0 < q < 1, the q-numerical range WqðAÞ ofA contains the range

Wq diag n2, n2π2� �� � ð14Þ

for any integer n ≥ 2: Let λ = n2π2 > 0: Then, the range

Wq diag π2, λ� �� �

= z = x + iy : x, yð Þ ∈ℝ2, x − q π2 + λ� �2

� �2(

+ y2

1 − q2≤

λ − π2

2

� �2)

ð15Þ

contains the convex polygon with vertexes

− 1 − qð Þ λ2 , 1 + qð Þ λ2 ,

−i 1 − q2� � λ

2 , i 1 − q2� � λ

2 :ð16Þ

For a sufficiently large λ, such a convex polygon containsan arbitrary given point in the Gaussian plane. Thus, this dif-ferential operator satisfies WqðAÞ =ℂ

(iii) If we restrict our attention for the variationalapproximation to the Schr€odinger operator, wemay have another story. The approximation Ak inTheorem 9 is given by a diagonal matrix with somereal eigenvalues fλ1, λ2,⋯,λkÞg with λ1 ≤ λ2 ≤⋯ ≤λk. The q-numerical range WqðAkÞ of Ak for 0 < q< 1 is given by WqðAkÞ = diag ðλ1, λkÞ. Then, therange Wqð diag ðλ1, λkÞÞ is the union of closed cir-cular discs

Wq diag λ1, λkð Þð Þ = z = a + ib : a, bð Þ ∈ℝ2, a −q λ1 + λkð Þ

2

� �2(

+ b2

1 − q2≤

λk − λ1ð Þ2

� �2)

ð17Þ

3.3. Application to Block Differential Operators. In this sub-section, we apply Theorem 9 to compute the q-numericalrange of Stokes-type operators and Hain-Lüst-type opera-tors. First, we study Stokes-type operators.

3.3.1. Application to Stokes-Type Operators. Consider the dif-ferential expression in the Hilbert space L22ð0, 1Þ≔ L2ð0, 1Þ⊕ L2ð0, 1Þ, we introduce the matrix differential operator

A0 ≔A0 B0

C0 D0

!=

−d2

dx2−ddx

ddx

−32

0BB@

1CCA, ð18Þ

on the domain

D A0ð Þ≔ H2 0, 1ð Þ ∩H1 0, 1ð Þ� �⊕H1 0, 1ð Þ: ð19Þ

The operator matrix A0 is not closed but closable [2]. Inorder to show that its closure is self-adjoint, we need thefollowing:

Remark 11.

(i) Consider the operator M given by

D Mð Þ≔y1

y2

!: y1 ∈H1

0 0, 1ð Þ, y1′ + y2 ∈H1 0, 1ð Þ,  

(

and y2 ∈ L2 0, 1ð Þ),

ð20Þ

such that

My1

y2

!=

− y′1 + y2

′

y′1 −12 y2

0B@

1CA: ð21Þ

Lemma 12. The operatorM in Equation ((21)) coincides withthe adjoint of the block operator matrix in Equation ((18)).

Proof. Suppose that

g =g1

g2

!∈D ~A

ð22Þ

4 Journal of Applied Mathematics

with

u =u1

u2

!∈ L2 0, 1ð Þ2: ð23Þ

Then

A0y, gð Þ = y, uð Þ, y =y1

y2

!∈D A0ð Þ

= H2 0, 1ð Þ ∩H1 0, 1ð Þ� �⊕H1 0, 1ð Þ

ð24Þ

or, equivalently,

− y′1 + y2

′, g1

L2 0,1ð Þ+ y′1 −

12 y2, g2

� �L2 0,1ð Þ

= y1, u1ð ÞL2 0,1ð Þ + y2, u2ð ÞL2 0,1ð Þ ð25Þ

for all y1, y2 ∈ C∞0 ð0, 1Þ with compact support contained in

ð0, 1Þ: In particular, when y1 = 0, then Equation (25)

becomes

−y2′ , g1

L2 0,1ð Þ+ −

12 y2, g2

� �L2 0,1ð Þ

= y2, u2ð ÞL2 0,1ð Þ: ð26Þ

The first part of the left-hand side of Equation (26) is abounded linear functional of y2, which means that g1 ∈H

1ð0, 1Þ, and also, the second part of the left-hand side of equa-tion (26) is a bounded linear functional of y2, which meansthat g2 ∈ L2ð0, 1Þ; again return to Equation (26), and inte-grating by part, we obtain

−y2, g1′ −12g2

� �= y2, u2ð Þ y2 ∈ C∞

0 0, 1ð Þ: ð27Þ

Because C∞0 ð0, 1Þ is a dense in L2ð0, 1Þ, this means that

u2 = g′1 − ð1/2Þg2: By the same argument, if we set y2 = 0y1 ∈H

10ð0, 1Þ ∩H2ð0, 1Þ, we find that

−y′′1, g1

L2 0,1ð Þ+ y1′ , g2

L2 0,1ð Þ= y1, u1ð ÞL2 0,1ð Þ: ð28Þ

Again, return to Equation (28), and integrating by part, we

Real axis Real axis

–30

–20

–10

0

10

20

30

Imag

inar

y ax

is

Numerical range

(𝜆) ≥ 0 (𝜆) = 0

Actual eigenvalues

10008006004002000–200–400

–300

–200

–100

0

100

200

300

400

Imag

inar

y ax

is

The q-numerical range of A.

1009080706050403020100

Figure 1: On the left-hand side, the variational method approximation to the Schrödinger operator. The approximationAk is given by a self-adjoint matrix with some real eigenvalues fλ1, λ2,⋯,λkg with λ1 ≤ λ2 ≤⋯λk: The q -numerical range WqðA24Þ of A24 for q = 1 is the linesegment ½λ1, λ24�; the red dots are σðA24Þ. On the right-hand side, the q -numerical range WqðA24Þ of A24 for 0 < q < 1 is the union of theclosed circular discs. The small red circles on the real axis inside the circular disk are σðA24Þ scaled by q.

5Journal of Applied Mathematics

obtain

−y1′ xð Þ �g1 xð Þh ix=1

x=0+ −y1,

�g1′ + g2

′� �

= y1, u1ð Þ: ð29Þ

The first part of the left-hand side of Equation (29) isbounded only when g1ð0Þ = 0 = g1ð1Þ, and the second partis bounded linear functional of y1, which means that g1′ +g2 ∈H

1ð0, 1Þ: Then, this implies that g ∈Dð ~AÞ, and becauseC∞0 ð0, 1Þ is a dense in L2ð0, 1Þ, this means that u1 = −

ðg1′ + g2Þ′ on ð0, 1Þ: It follows that

u1

u2

!= ~A

g1

g2

!: ð30Þ

Remark 13.

(i) By the same argument of Lemma 12, it is not difficultto see that the operator A0 in Equation (18) is sym-metric in L2ð0, 1Þ2 with the domain

D Mð Þ≔y1

y2

!: y1 ∈H1

0 0, 1ð Þ, y1′ + y2 ∈H1 0, 1ð Þ, and y2 ∈ L2 0, 1ð Þ

( ),

My1

y2

!=

− y′1 + y2

′

y′1 −12 y2

0B@

1CA

ð31Þ

(ii) Because M is a symmetric operator and has non-empty resolvent, then M self-adjoint. Thus, by [10,Theorem 5.4],M = �A0

The following result shows that the eigenvalues of theStocks operator coincide with eigenvalues of the operatorM.

Proposition 14. Let A0 be as in Equation((18)), then σpð �A0

Þ = σpðA0Þ:

Proof. Let λ ∈ σpð �A0Þ, then there exists an y = ðy1 y2Þt ∈Dð�A0Þ, y ≠ 0, such that

�A0y1

y2

!= λ

y1

y2

!: ð32Þ

We see that Equation (32) is equivalent to the followingsystem of equations:

− y1′ + y2

′ = λy1, ð33Þ

y′1 −12 y2 = λy2, ð34Þ

where ðy1 y2Þt ∈Dð �A0Þ and

D �A0� �

≔y1

y2

!: y1 ∈H1

0 0, 1ð Þ, y1′ + y2 ∈H1 0, 1ð Þ,  

(

and y2 ∈ L2 0, 1ð Þ):

ð35Þ

Because y1 ∈H1ð0, 1Þ, then from Equation (33), we get

y′1 + y2 ∈H2ð0, 1Þ and Equation (34) can be written as

y′1 + y2 = λ + 32

� �y2 ; ð36Þ

this implies that y2 ∈H2ð0, 1Þ for λ ≠ ð−3/2Þ: Since H2ð0, 1Þ

is a linear space then y′1 ∈H2ð0, 1Þ, hence y1 ∈H3ð0, 1Þ ⊆

H2ð0, 1Þ, it follows that y = ðy1, y2Þt ∈DðA0Þ, so λ is aneigenvalue of A0:

Conversely, if λ ∈ σpðAÞ, then there exists an y =ðy1, y2Þt ∈DðA0Þ ⊆Dð �A0Þ,y ≠ 0, such that

A0y1

y2

!= λ

y1

y2

!: ð37Þ

This means that y is an eigenfunction of �A0 since�A0jDðA0Þ =A0:

Remark 15.

(i) It may be shown that A has two series of eigenvaluesλ±k given by

λ±k =k2π2 − 3/2 ±

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3/2 − k2π2� �2 + 10k2π2

q2 , k = 1, 2,⋯:

ð38Þ

The eigenvalues of the minus series are located in theinterval (-5/2,-3/2] and convergent to -5/2 as k⟶∞: Theeigenvalues of the plus series are located in the interval½0,∞Þ and convergent to ∞ as k⟶∞: The essentialspectrum of A is the accumulation point of the eigen-values: σessðMÞ = f−5/2g

(ii) It is not difficult to see that the subspace C0 ≔CA⊕CD ⊂DðA0Þ = ðDðA0Þ ∩DðC0ÞÞ ⊕ ðDðB0Þ ∩DðD0ÞÞ is a core of A0

(iii) Now form the matrix elements h−ϕk′′ , ϕji, h−ϕk′ , ϕ ji,hϕk′ , ϕji, h−ð3/2Þϕk, ϕji using the inner product in(6) with respect to the orthonormal basis in Equa-tion (12) and consider the (infinite) block operatormatrix

6 Journal of Applied Mathematics

G ≔

−ϕk′′ , ϕj

D E−ϕk′ , ϕj

D E

ϕk′ , ϕj

D E−32 ϕk, ϕj

� �0BBBB@

1CCCCA: ð39Þ

The defined matrix Ak in (4) is obtained by takingthe leading submatrices of the block G , with appropriatedimensions. Observe that h−ϕk′′ , ϕji = diag fπ2, 4π2, 9π2,⋯g,

−ϕk′ , ϕj

D E=

0, ifk = j ;

−2kπð10cos kπxð Þ  sin jπxð Þdx, ifk ≠ j ;

0B@

ϕk′ , ϕj

D E=

0, ifk = j ;

2kπð10cos kπxð Þ  sin jπxð Þdx, ifk ≠ j ;

0B@

ð40Þ

and h−ð3/2Þϕk, ϕji = −ð3/2Þ δk,j, which can be evaluatedexplicitly

Figure 2 shows attempts to calculate WqðAkÞ for variousk and q and also some attempts to estimate these sets by qual-itative means, using existing theorems from the literature aswell as the theorems proved above.

3.4. Analytical Estimates for Stokes Operator. In order tounderstand the results in Figure 2, it is useful to find an ana-

lytical estimate for WqðMÞ. Let ζ!, η! ∈DðMÞ, where

ζ!=

ζ1

ζ2

!,

η! =

η1

η2

! ð41Þ

with ∥ζ!∥ = ∥η!∥ = 1, hζ!, η!i = q and let

λ = M ζ!, η!

� �= −D2ζ1, η1 �

+ −ζ2′ , η1D E

+ ζ1′ , η2D E

+ αζ2, η2h i,

ð42Þ

where α = −ð3/2Þ: Equation (42) gives, as an estimate forthe first term on the right-hand side of (42),

ð10− ζ1′′�η1dx =

ð10ζ1′�η1′dx ≥ qπ2

ð10ζ1�η1 dx: ð43Þ

For the second and the third term on the right-hand side

of (42), the Cauchy-Schwarz inequality and Youngs inequal-ity yield

Reð10− ζ2 ′�η1 dx

� �= Re

ð10ζ2�η1 ′ dx

� �≥ −ð10

ζ2j j2 + η1 ′�� ��2 dx

,

ð44Þ

Reð10ζ1 ′�η2 dx

� �≥ −ð10

ζ1 ′�� ��2 + η2j j2 dx

: ð45Þ

Combining Equations (44) and (45), we get that

Reð10ζ2 ′�η1 dx

� �+ Re

ð10ζ1 ′�η2 dx

� �≥ −q: ð46Þ

The fourth term of the right-hand side of Equation (42)satisfies

Reð10αζ2�η2 dx

� �≥ inf Re αð Þ q −

ð10ζ1�η1 dx

� �: ð47Þ

Hence, from Equations (43), (46), and (47), we get that

Re λð Þ ≥ qπ2ð10ζ1�η1dx − q + inf Re αð Þ q −

ð10ζ1�η1 dx

� �:

ð48Þ

This simplifies to

Re λð Þ ≥ qπ2ð10ζ1�η1 dx − q + q −

ð10ζ1�η1

� �inf Re αð Þ

= q π2 − inf R αð Þ� �ð10ζ1�η1dx + q inf Re αð Þ − 1ð Þ:

ð49Þ

This yields

Re λð Þ ≥q α − 1ð Þ, if π2 − α ≥  0 ;q π2 − 1� �

, if π2 − α <  0:

(ð50Þ

For our example, these yield Re ðλÞ ≥ −ð5/2Þq:To estimate Im ðλÞ, observe that

Im λð Þ =ð10Im uð Þð Þζ2�η2 dx = 0: ð51Þ

This completes the estimates on WqðMÞ.3.5. The Hain-Lüst Operator. This operator was introducedby Hain and Lüst in application to problems of magnetohy-drodynamics [11], and the problems of this type were studiedin [7, 8, 12]. Assume that w : ½0, 1�→ ½0,∞Þ, ~w : ½0, 1�→ ½0,∞Þ, and u : ½0, 1�→ℂ are such that wðxÞ = 1, ~wðxÞ = 1, uðxÞ = 18e2πix − 20, for each x ∈ ½0, 1�. We introduce the

7Journal of Applied Mathematics

differential expression

τ~A ≔ −d2

dx2,

τ~B ≔w xð Þ,τ~C ≔ ~w xð Þ,τ~D ≔ u xð Þ:

ð52Þ

Let A, B, C,D be the operators in the Hilbert space L2ð0, 1Þ induced by the differential expressions τ~A, τ~B, τ~C , andτ~D with domain

D Að Þ≔H2 0, 1ð Þ ∩H10 0, 1ð Þ,

D Bð Þ =D Cð Þ =D Dð Þ≔ L2 0, 1ð Þ:ð53Þ

In the Hilbert space L22ð0, 1Þ≔ L2ð0, 1Þ ⊕ L2ð0, 1Þ, weintroduce the matrix differential operator

A ≔A B

C D

!= −

d2

dx2w xð Þ

~w xð Þ u xð Þ

0B@

1CA, ð54Þ

on the domain

D Að Þ≔ H2 0, 1ð Þ ∩H10 0, 1ð Þ� �

⊕ L2 0, 1ð Þ: ð55Þ

Remark 16.

(i) By [13], Corollary VII.2.7, the operator A≔ −ðd2/dx2Þ with domain DðAÞ≔ ðH2ð0, 1Þ ∩H1

0ð0, 1ÞÞ isclosed. Moreover, because DðAÞ ⊂DðCÞ, then theoperator C is A-bounded with relative bound 0. Thisfollows since there is a γ > 0 such that, for every ε >0,

∥Cf ∥2 ≤ γ∥ Af , fh i∥ ≤ γ ε∥Af ∥2 + ε−1∥f ∥2� �

: ð56Þ

On the other handDðDÞ ⊂DðBÞ, then the operator B isD-bounded, we conclude that the operator matrix DðAÞ isdiagonally dominant of order 0; it is closed by [14], Corollary2.2.9 (i)

(ii) In Equation (54), since A is self-adjoint with thepurely discrete spectrum, then the linear span CA= span fϕ1, ϕ2,⋯g is a core of A, where fϕk : k ∈ℕg is an orthonormal basis in L2ð0, 1Þ, and by thesame argument because D in Equation (54) is

Real axis

–200

–150

–100

–50

0

50

100

150

200

Imag

inar

y ax

isClassical numerical range of A

500400300200100–100 0

(𝜆) > –2.5 (𝜆) = 0

–200

–150

–100

–50

0

50

100

150

200

Imag

inar

y ax

isReal axis

5004003002001000–100

Approximation of the q-numerical range of M.

Figure 2: On the left-hand side, the variational method approximation to the Stokes operator. The approximation Ak is given by a realsymmetric matrix with some eigenvalues fλ1, λ2,⋯,λ28g with λ1 ≤ λ2 ≤⋯λ28: The q -numerical range WqðA28Þ of A28 for q = 1 is the linesegment ½λ1, λ28�; the red dots are σðA28Þ. On the right-hand side, the q-numerical range WqðA28Þ of A28 for 0 < q < 1 is the union of

closed circular disk center at qx∗A28x with radiusffiffiffiffiffiffiffiffiffiffiffiffi1 − q2

prðxÞ where rðxÞ =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi∥A28x∥2 − jx∗A28xj2

q: The small red circles inside the

circular disk are σðA28Þ scaled by q.

8 Journal of Applied Mathematics

bounded, then the linear span CD = span fϕ1, ϕ2,⋯g is a core of D:Hence, it is not difficult to see thatthe subspace C ≔CA ⊕CD ⊂DðAÞ = ðDðAÞ ∩DðCÞÞ ⊕ ðDðBÞ ∩DðDÞÞ is a core of A . So the mainTheorem 9 is applicable to this example

(iii) We may use the eigenfunctions in Equation (12) asbasis elements for a discretization of the type dis-cussed in Section 2, forming the matrix elements hAϕk, ϕji, hwϕk, ϕji, h~wϕk, ϕji, huϕk, ϕji, with respectto the inner product in (6) and considering the infi-nite block matrix

Q≔

Aϕk, ϕj

D Ewϕk, ϕj

D E

~wϕk, ϕj

D Euϕk, ϕj

D E0BBB@

1CCCA ð57Þ

The defined matrix Ak in (4) is obtained by taking theleading submatrices of the block Q, with appropriatedimensions.

Observe that hAϕk, ϕji = diag fπ2, 4π2, 9π2,⋯g,hwϕk, ϕji = diag f1, 1, 1,⋯g, h~wϕk, ϕji = diag f1, 1, 1,⋯g,and huϕk, ϕji = 36

Ð 10 e

2πix sin ðkπxÞ sin ðjπxÞdx − 20δk,j,which can be evaluated explicitly. If the operator A includeda potential, for instance, then its eigenfunctions would notgenerally be explicitly computable. We could still use thefunctions ϕj in (12) as basis functions, but the matrix ele-ments hAϕk, ϕji would have to be computed by quadratureand the corresponding matrix would no longer diagonal.Figure 3 shows attempts to compute the numerical approxi-mation of the boundary of WqðAkÞ for various k and q andalso some attempts to estimate these sets by qualitativemeans, using existing theorems from the literature as wellas the theorems proved above.

3.6. Analytical Estimates for Non-Self-Adjoint Hain-LüstOperator. In order to understand the results in Figure 3, itis useful to find an analytical estimate for WqðAÞ. Let x!, y!∈DðAÞ, where

x! =x1

x2

!,

y! =y1

y2

! ð58Þ

with ∥x!∥ = ∥y!∥ = 1, ðx!, y!Þ = q and let

λ = A x!, y!D E

= −D2x1, y1 �

+ x2, y1h i + x1, y2h i + ux2, y2h i:ð59Þ

The first term of (59) gives, as an estimate,

ð10x1 ′′�y1 dx ≥ qπ2

ð10x1�y1 dx: ð60Þ

For the second and the third term on the right-hand sideof (59), the Cauchy-Schwarz inequality and Youngs inequal-ity yield

R

ð10x2�y1 dx

� �≥ −ð10

x2j j2 + y1j j2� �dx, ð61Þ

R

ð10x1�y2 dx

� �≥ −ð10

x1j j2 + y2j j2� �dx: ð62Þ

Combining Equations (61) and (62), we get that

R

ð10x2�y1 dx

� �+R

ð10x1�y2 dx

� �≥ −q: ð63Þ

The fourth term of the right-hand side of Equation (59)satisfies

R

ð10ux2�y2 dx

� �≥ inf  R uð Þ q −

ð10x1�y1 dx

� �: ð64Þ

Hence, from Equations (60), (63), and (64), we get that

R λð Þ ≥ qπ2ð10x1�y1dx − q + inf  R uð Þ q −

ð10x1�y1 dx

� �:

ð65Þ

This simplifies to

R λð Þ ≥ qπ2ð10x1�y1 − q + q −

ð10x1�y1

� �inf  R uð Þ

= q π2 − inf  R uð Þ� �ð10x1�y1dx + q inf  R uð Þ − 1ð Þ:

ð66Þ

This yields

R λð Þ ≥q inf  R uð Þ − 1ð Þ, if π2 − inf R uð Þ ≥  0 ;q π2 − 1� �

, if π2 − inf R uð Þ <  0:

(

ð67Þ

For our example, these yield RðλÞ ≥ −53q:

9Journal of Applied Mathematics

To estimate Im ðλÞ, observe that

Im λð Þ =ð10Im uð Þð Þx2�y2 dx ≤ sup

x∈ 0,1½ �Im uð Þð Þ

ð10x1�y1 dx ≤ q25,

Im λð Þ =ð10Im uð Þð Þx2�y2 dx ≥ inf

x∈ 0,1½ �Im zð Þð Þ

ð10x2�y2 dx ≥ −q25,

ð68Þ

hence

−q25 ≤ Im λð Þ ≤ q25: ð69Þ

This completes the estimates on WqðAÞ.The following result shows that the q -numerical range of

Example 1, is unbounded.

Remark 17. Suppose that

x! =ffiffiffi2

psin kπxð Þ0

!,

y! =ffiffiffi2

psin kπxð Þ0

! ð70Þ

where x!, y! ∈DðAÞ, with ∥x!∥ = ∥y!∥ = 1, and ðx!, y!Þ = q then

Ax!, y!D E

=ffiffiffi2

p 2kπð Þ2

ð10sin2 kπxð Þdx = k2π2: ð71Þ

But k ∈ℤ can be arbitrarily large. This means that RðλÞ isunbounded above.

3.6.1. Application to Self-Adjoint Hain-Lüst Operator. In theHilbert space H = L2ð−π, πÞ ⊕ L2ð−π, πÞ, we introduce thematrix differential operator

~A ≔L B

C V

!= −

d2

dx2+ 4 1

1 z

0@

1A, ð72Þ

where

z xð Þ =0, for − π ≤ x < 0,2, for 0 ≤ x < π,

(ð73Þ

on the domain

D ~A

≔ H2 −π, πð Þ ∩H10 −π, πð Þ� �

⊕ L2 −π, πð Þ: ð74Þ

Remark 18.

Real axis

Imag

inar

y ax

is

–60

–40

–20

0

20

40

60

q = 1

Imag

inar

y ax

is–60

–40

–20

0

20

40

60

100500–50–100

(𝜆) > –53 (𝜆) < 25–25 <

q = 0.7 (𝜆) > –37.1

(𝜆) < 17.5–17.5 <

q-numerical range of A q-numerical range of A

Real axis10080600 20 40–20–40–60–80–100

Figure 3: On the left-hand side, the variational method approximation to the Hain-Lüst operator for computing q -numerical rangeWqðA48ÞofA48 for q = 1. The red dots are σðA48Þ. On the right-hand side, for 0 < q < 1, the variational method approximation for computingWqðA48Þof A48 for 0 < q < 1. The red dots are σðA48Þ scaled by q.

10 Journal of Applied Mathematics

(i) It is not difficult to see that the subspace C ≔CA ⊕CV ⊂Dð ~AÞ = ðDðLÞ ∩DðCÞÞ ⊕ ðDðBÞ ∩DðVÞÞ is acore of ~A . So the main Theorem 9 is applicable to thisexample

(ii) The eigenvalues and normalized eigenfunctions ofthe operator −d2/dx2 in L2ð−π, piÞ are

λj =j2

4 ,

ψj xð Þ = sin j x + πð Þ/2ð Þffiffiffiπ

p  for j = 1, 2, 3,⋯ð75Þ

Now form the matrix elements hAψk, ψji, hBψk, ψji, hCψk, ψji, hDψk, ψji, with respect to orthonormal basis ψjðxÞ= ð1/ ffiffiffi

πp Þ  sin ðjðx + πÞ/2Þ where j ∈ℕ and consider the

infinite block operator matrix

P ≔Aψk, ψj

Bψk, ψj

Cψk, ψj

Dψk, ψj

0B@

1CA: ð76Þ

The defined matrix Ak in (4) is obtained by taking theleading submatrices of the block P , with appropriate

dimensions. Observe that hLψk, ψji = diag f1/4, 1/2,⋯g,hBψk, ψji = hCψk, ψji = diag f1, 1, 1,⋯g, and

Vψk, ψj

D E=

1π

ðπ0sin j x + πð Þ

2

� �sin k x + πð Þ

2

� �, if k ≠ j ;

12 , if k = j ;

0BB@

ð77Þ

which can be evaluated explicitly. Figure 4 shows attempts tocalculateWqðAkÞ for various k and q and also some attemptsto estimate these sets by qualitative means, using existing the-orems from the literature as well as the theorems provedabove.

3.7. Analytical Estimates for Self-Adjoint Hain-Lüst Operator.In order to understand the results in Figure 4, it is useful tofind an analytical estimate for Wqð ~AÞ. By the same argu-ments in Section 3.6, it is not difficult to see that Re ðλÞ ≥−ð3/4Þq, and

Im λð Þ =ðπ−π

Im zð Þð Þx2 �y2 dxÞ = 0: ð78Þ

Imag

inar

y ax

is

–1

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

1q-numerical range of A

Real axis

–5

–4

–3

–2

–1

0

1

2

3

4

5

Imag

inar

y ax

is

Approximation of the q-numerical range of M

0 0.2 0.4 0.6 0.8 1–0.2–0.4–0.6 –4 –2 0 2 4 6 8 10 12–0.8–1Real axis

q = 1 (𝜆) > –0.7500 (𝜆) = 0

Figure 4: On the left-hand side, the variational method approximation to the Hain-Lüst operator. The approximation Ak is given by a realsymmetric matrix with some eigenvalues fλ1, λ2,⋯,λ28g with λ1 ≤ λ2 ≤⋯λ28: The q -numerical range WqðA28Þ of A28 for q = 1 is the linesegment ½λ1, λ28�; the red dots are σðA28Þ. On the right-hand side, the q -numerical range WqðA28Þ of A28 for 0 < q < 1 is the union of

closed circular disk center at qx∗A28x with radiusffiffiffiffiffiffiffiffiffiffiffiffi1 − q2

prðxÞ where rðxÞ =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi∥A28x∥2 − jx∗A28xj2

q: The small red circles inside the

circular disk are σðA28Þ scaled by q.

11Journal of Applied Mathematics

4. Conclusions

This paper illustrates the practical difficulties associated withthe computation of q -numerical ranges of operator matricesand block operator matrices of differential operators, evenwhen good theoretical results are available to underpin theapproximation procedure.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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12 Journal of Applied Mathematics