Nonlinear Analysis 126 (2015) 131–142

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Nonlinear Analysis

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Linear Algebra and its Applications 466 (2015) 102–116

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Linear Algebra and its Applications

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Inverse eigenvalue problem of Jacobi matrix

with mixed data

Ying Wei 1

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China

a r t i c l e i n f o a b s t r a c t

Article history:Received 16 January 2014Accepted 20 September 2014Available online 22 October 2014Submitted by Y. Wei

MSC:15A1815A57

Keywords:Jacobi matrixEigenvalueInverse problemSubmatrix

In this paper, the inverse eigenvalue problem of reconstructing a Jacobi matrix from its eigenvalues, its leading principal submatrix and part of the eigenvalues of its submatrix is considered. The necessary and sufficient conditions for the existence and uniqueness of the solution are derived. Furthermore, a numerical algorithm and some numerical examples are given.

© 2014 Published by Elsevier Inc.

E-mail address: weiyingb@gmail.com.1 Tel.: +86 13914485239.

http://dx.doi.org/10.1016/j.laa.2014.09.0310024-3795/© 2014 Published by Elsevier Inc.

Subelliptic Peter–Weyl and Plancherel theorems on compact,connected, semisimple Lie groupsAndras DomokosDepartment of Mathematics and Statistics, California State University at Sacramento, 6000 J Street,Sacramento, CA, 95819, USA

a r t i c l e i n f o

Article history:Received 3 December 2014Accepted 1 April 2015Communicated by Enzo Mitidieri

Keywords:Semisimple Lie groupsSub-Riemannian geometrySubelliptic Laplacian

a b s t r a c t

We will study the connections between the elliptic and subelliptic versions of thePeter–Weyl and Plancherel theorems, in the case when the sub-Riemannian struc-ture is generated naturally by the choice of a Cartan subalgebra. Along the way wewill introduce and study the subelliptic Casimir operator associated to the subellip-tic Laplacian.

© 2015 Elsevier Ltd. All rights reserved.

1. Introduction

In this paper we will show that the subelliptic spectral analysis results needed to study the subellipticheat kernel and to obtain precise bounds of the L2-norms of the non-horizontal directional derivatives, havenatural and simple expressions in the case of compact and semi-simple Lie groups. The key ingredient is theconnection between the elliptic and subelliptic versions of the Peter–Weyl and Plancherel theorems, whenthe subelliptic structure is generated by the orthogonal complement of a maximal commutative subalgebraof the Lie algebra.Studies regarding the subelliptic heat kernel started immediately after Hormander’s paper [13] appeared.Initially these studies were focused mostly on the nilpotent Lie groups [12], with a more recent shifttoward non-nilpotent Lie groups [1–5], followed independently, and somewhat logically, by parallel studieson regularity of nonlinear subelliptic PDE’s [7,8,6].

Let G be a compact, connected, semi-simple matrix Lie group and G its Lie algebra. In this context,working with matrix groups is not a restriction, but has the advantage of an easy setup for the formulas wewill use. We denote byMn(R) orMn(C) the vector space of n×n real or complex matrices and by GLn(R)or GLn(C) the Lie groups of the invertible n×n matrices. Matrix groups are defined as closed subgroups ofGLn(R) or GLn(C) and hence inherit the Lie group structure.

E-mail address: domokos@csus.edu.

http://dx.doi.org/10.1016/j.na.2015.04.0100362-546X/© 2015 Elsevier Ltd. All rights reserved.

132 A. Domokos / Nonlinear Analysis 126 (2015) 131–142

The Lie algebra of G can be defined by using the matrix exponential:

G = X ∈Mn(R or C depending on G) | exp(tX) ∈ G, ∀ t ∈ R .

From this definition it follows that G is a real vector space and contains the generators of the 1-parametersubgroups of G. G becomes an algebra when it is endowed with the bilinear operator,

[X,Y ] = XY − Y X,

called the commutator of X and Y , which measures the non-commutativity at infinitesimal scales in G. Acommutative Lie algebra has the property that [X,Y ] = 0, ∀ X,Y ∈ G and a semi-simple Lie algebra is onthe opposite end of the commutativity scale, as it cannot have any non-trivial commutative ideals. A Liegroup is semi-simple if its Lie algebra is semi-simple.

The adjoint representation of G is the group homomorphism

Ad : G→ Aut(G), Adx (X) = xXx−1,

while its differential at the identity is the Lie algebra homomorphism

ad : G → End(G), adX (Y ) = [X,Y ].

The Killing form

K(X,Y ) = trace (adX · adY ),

is negative definite and non-degenerate on the Lie algebra of a compact, semi-simple Lie group, and hencewe can define an inner product on G as

⟨X,Y ⟩ = −ρK(X,Y ), (1.1)

where ρ > 0 is a constant, which can be adjusted for each Lie algebra according to our normalization needs.The Killing form is Ad invariant, therefore Adx is a unitary linear transformation for all x ∈ G and adX isskew-symmetric for all X ∈ G.

We will consider a natural sub-Riemannian geometry on G, which is defined by the choice of a maximal,commutative subalgebra of G, called a Cartan subalgebra.Let us fix a Cartan subalgebra Γ and let T = T1, . . . , Tr be an orthonormal basis of it.We extend the inner product of G bi-linearly to the complexified Lie algebra GC = G

iG. The mappings

adT : GC → GC, T ∈ Γ , commute and are skew-symmetric, so they can be simultaneously diagonalized andhave purely imaginary eigenvalues.We define α ∈ Γ to be a root if α = 0 and Gα = 0, where

Gα = Z ∈ GC : adT (Z) = i ⟨α, T ⟩Z, ∀ T ∈ T . (1.2)

Let R be the set of all roots, which will be ordered by the relation α > β if α − β has its first non-zerocoordinate positive. We denote by P the set of all positive roots and let

δ = 12α∈P

α.

For the most important properties of Gα we quote [9,14]:

(i) dimC Gα = 1.(ii) G0 = ΓC.

(iii) G−α = Gα.(iv) ⟨Gα,Gβ⟩ = 0 if β = ±α.

A. Domokos / Nonlinear Analysis 126 (2015) 131–142 133

(v) [Gα,Gβ ]

= Gα+β if α+ β ∈ R= 0 if α+ β ∈ R and α+ β = 0⊂ iT if α+ β = 0.

(vi) If Eα ∈ Gα and E−α ∈ G−α then [Eα, E−α] = i ⟨Eα, E−α⟩ α.

The above properties of Gα and the real root space decomposition

G = T ⊕α∈P

Re (Gα ⊕ G−α),

allow us to choose an orthonormal basis of Γ⊥,

X = X1, . . . , Xk, Y1, . . . , Yk, (1.3)

with the following properties:

Proposition 1.1.

(i) For all 1 ≤ j ≤ k there exists αj ∈ P such that Xj , Yj ∈ Re(Gαj ⊕ G−αj )(ii) E±αj = Xj ± Yj ∈ G±αj .(iii) ⟨Eαj , E−αj ⟩ = 2.(iv) [Xj , Yj ] = −αj .(v) Γ ⊂ [X ,X ].

The formula

Xf(x) = d

dt

t=0

f(x · exp tX) (1.4)

identifies each X ∈ G with a left invariant vector field acting on smooth functions. By property (v) ofProposition 1.1, Γ⊥ defines the horizontal distribution of a sub-Riemannian geometry on G. The subellipticLaplacian corresponding to this sub-Riemannian geometry is

∆X =kj=1

X2j + Y 2

j ,

while the elliptic Laplacian is

∆ =kj=1

X2j + Y 2

j +rm=1

T 2m.

Example 1.1 ([6,14]). An example for a compact, semi-simple Lie group is the special orthogonal group

G = SO(n) = x ∈ GLn(R) : xxT = I, detx = 1,

which has the Lie algebra,

G = so(n) = X ∈Mn(R) : X +XT = 0.

A basis of so(n) is given by the matrices Xij for all 1 ≤ i < j ≤ n, having all entries 0 except the ijth entrywhich is 1 and the jith entry which is −1.

In so(3) we can use ρ = 12 in the inner product (1.1) and choose

T = X12, α = 1, X1 = X13, Y1 = X23.

134 A. Domokos / Nonlinear Analysis 126 (2015) 131–142

In so(4) we can use ρ = 14 in the inner product (1.1) and choose

T1 = X12, T2 = X34,

α1 = X12 −X34, α2 = X12 +X34,

X1 = 1√2

(X13 +X24), Y1 = 1√2

(X23 −X14),

X2 = 1√2

(X13 −X24), Y2 = 1√2

(X23 +X14).

2. The subelliptic Peter–Weyl and Plancherel theorems

In this section we prove the subelliptic versions of the classical Peter–Weyl and Plancherel theorems,which are related to the spectral analysis of the subelliptic Laplacian.Let G be the set of equivalence classes of irreducible, unitary representations of G. If [π] ∈ G, thenπ : G→ AutV π is a group homomorphism and V π is a finite dimensional complex vector space of dimensiondπ. The derived representation of π is defined as

dπ : G → End(V π), dπ(X)v = d

dt

t=0

π(etX)v.

dπ is a Lie algebra homomorphism, as the differential at the identity of a smooth group homomorphism.The entry functions of the representation π are defined for every u, v ∈ V π by

πu,v : G→ C, πu,v(x) = ⟨π(x)u, v⟩.

In our case of a compact, semi-simple Lie group, these entry functions are smooth and their directionalderivatives in the directions of the flows of left invariant vector fields are calculated by

Xπu,v(x) = d

dt

t=0

πu,v(xetX) = d

dt

t=0⟨π(xetX)u, v⟩

= d

dt

t=0⟨π(x) π(etX)u, v⟩ = ⟨π(x) dπ(X)u, v⟩. (2.1)

As G is compact, we can fix a normalized bi-invariant Haar measure µ, and the entry functions πu,v belongto L2(G). Therefore, we can define the subspace of L2(G) generated by the entry functions of π,

Eπ = span πu,v : u, v ∈ V π ⊂ L2(G),

which depends only on the equivalence class of π.We recall now Schur’s orthogonality relations [10,11]:

Theorem 2.1. Let π be an irreducible, unitary representation of G. Then for all u1, u2, v1, v2 ∈ V π we haveGπu1,v1(x) · πu2,v2(x) dµ(x) = 1

dπ⟨u1, u2⟩ · ⟨v1, v2⟩. (2.2)

Theorem 2.2. Let π1 and π2 be irreducible, unitary representations of G such that [π1] = [π2]. Then for allu1, v1 ∈ V π

1 and u2, v2 ∈ V π2 we have

Gπ1u1,v1(x) · π2

u2,v2(x) dµ(x) = 0. (2.3)

The following theorem is fundamental for harmonic analysis on compact groups:

A. Domokos / Nonlinear Analysis 126 (2015) 131–142 135

Theorem 2.3 (The Peter–Weyl Theorem).

L2(G) =[π]∈G

Eπ, (2.4)

where

means the closure of all finite linear combinations.

Corollary 2.1. If we choose an orthonormal basis eπj , 1 ≤ j ≤ dπ of each V π and defineπjk(x) = ⟨π(x)eπj , eπk ⟩, then √

dπ πjk : [π] ∈ G, 1 ≤ j, k ≤ dπ

is a complete orthonormal set in L2(G) and therefore for all f ∈ L2(G)

f(x) =

[π]∈G

dπj,k=1

cπjk πjk(x), (2.5)

where

cπjk = dπ

Gf(x) πjk(x) dµ(x).

Definition 2.1. (a) The Casimir operator associated to the orthonormal basis XT of G is

Ωπ : V π → V π, Ωπ =kj=1

dπ(Xj)2 + dπ(Yj)2 +rm=1

dπ(Tm)2.

(b) The subelliptic Casimir operator associated to X is defined as

ΩπX : V π → V π, ΩπX =kj=1

dπ(Xj)2 + dπ(Yj)2.

We can observe that, as in the case of the adjoint representation, the linear mappings dπ(T ), T ∈ Γ ,commute and are skew symmetric, therefore they share eigenspaces and have purely imaginary eigenvalues.Hence, for λ ∈ Γ we can define

V πλ = v ∈ V : dπ(T )v = i⟨λ, T ⟩v, ∀ T ∈ Γ,

and call λ ∈ Γ a weight of the representation dπ if V πλ = 0. Let us denote by Λπ the set of weights.On Λπ we use the same ordering as on R and let wπ be the highest weight. The following proposition is awell-established fundamental result:

Proposition 2.1 ([16]). For all u, v ∈ V π we have

Ωπu = −c2u,

and

∆πu,v = −c2 πu,v,

where

c2 = ∥wπ∥2 + 2⟨wπ, δ⟩.

136 A. Domokos / Nonlinear Analysis 126 (2015) 131–142

Proof. In order to see the connection between Proposition 2.1 and its subelliptic counterpart, Proposition 2.2,we shortly explain the main ideas of the proof. The first identity is a consequence of Schur’s lemma and itfollows from the facts that dπ(X) is skew symmetric, dπ is irreducible and Ωπ commutes with dπ.

The second identity can be proved by using (2.1) to get that for any X ∈ G we have

X2πu,v(x) = ⟨π(x) dπ(X)2u, v⟩,

and subsequently,

∆πu,v(x) = ⟨π(x) Ωπu, v⟩ = −c2πu,v(x).

To estimate c, notice that if α ∈ P, and Eα ∈ Gα then

dπ(Eα)(V πwπ ) = 0.

For all u ∈ V πwπ and v ∈ V π we have

Eα πu,v(x) = ⟨π(x) dπ(Eα)u, v⟩ = 0,

and hencekj=1

X2j + Y 2

j

πu,v = 1

2

kj=1

EαjE−αj + E−αjEαj

πu,v

= 12

kj=1

Eαj , E−αj

πu,v + 2E−αjEαjπu,v

=α∈P

iαπu,v =α∈P−⟨wπ, α⟩πu,v

= −2⟨wπ, δ⟩πu,v.

Moreover,

T 2mπu,v(x) = ⟨π(x) dπ(Tm)2u, v⟩ = −⟨wπ, Tm⟩2πu,v(x),

which implies thatrm=1

T 2mπu,v = −

rm=1⟨wπ, Tm⟩2πu,v = −∥wπ∥2πu,v.

In conclusion, for u ∈ V πwπ and v ∈ V π we have

∆πu,v =kj=1

X2j + Y 2

j

πu,v +

rm=1

T 2mπu,v

= −∥wπ∥2 + 2⟨wπ, δ⟩

πu,v,

which proves that c2 = ∥wπ∥2 + 2⟨wπ, δ⟩.

Proposition 2.1 shows that the decomposition of L2(G) in the Peter–Weyl theorem gives the spectraldecomposition of the elliptic Laplacian.We propose to get a similar result for the subelliptic Laplacian. As a first note, the subelliptic Casimiroperator ΩπX does not commute with dπ. Associated to the eigenspaces V πλ there is a orthogonal direct spacedecomposition

V π =λ∈Λπ

V πλ . (2.6)

A. Domokos / Nonlinear Analysis 126 (2015) 131–142 137

Proposition 2.2. For all u ∈ V πλ and v ∈ V π we have

ΩπXu = −∥wπ∥2 + 2⟨wπ, δ⟩ − ∥λ∥2

u,

and

∆Xπu,v = −∥wπ∥2 + 2⟨wπ, δ⟩ − ∥λ∥2

πu,v.

Proof. Let us consider T ∈ Γ , u ∈ V πλ and v ∈ V π. Then

dπ(T )2u = −⟨λ, T ⟩2u,

and hence

ΩπXu = Ωπu−rm=1

dπ(Tm)2u = Ωπu+rm=1⟨λ, Tm⟩2u

= −∥wπ∥2 + 2⟨wπ, δ⟩ − ∥λ∥2

u.

Therefore, for all x ∈ G we have

∆Xπu,v(x) = ⟨π(x) ΩπXu, v⟩ = −∥wπ∥2 + 2⟨wπ, δ⟩ − ∥λ∥2

πu,v(x).

Similarly to Eπ = spanπu,v : u, v ∈ V π, for λ ∈ Λπ let us define

Eπλ = spanπu,v : u ∈ V πλ , v ∈ V π.

As an immediate consequence of Propositions 2.1 and 2.2 we have the following corollary.

Corollary 2.2. (a) For all f ∈ Eπ we have

∆f = −(∥wπ∥2 + 2⟨wπ, δ⟩)f.

(b) For all f ∈ Eπλ we have

∆X f = −∥wπ∥2 + 2⟨wπ, δ⟩ − ∥λ∥2

f.

For each V πλ , λ ∈ Λπ, let us choose on orthonormal basis eλj and define the functions

πλjk(x) = ⟨π(x)eλj , eλk⟩.

Now, as Corollary 2.2 gives the eigenspaces of the subelliptic Laplacian, everything is prepared to state andprove the subelliptic version of Theorem 2.3. We use dπλ to denote the dimension of V πλ .

Theorem 2.4 (The Subelliptic Peter–Weyl Theorem).

L2(G) =[π]∈G

λ∈Λπ

Eπλ , (2.7)

and hence for all f ∈ L2(G) we have

f(x) =

[π]∈G

λ∈Λπ

dπλj,k=1

cπ,λjk πλjk(x), (2.8)

138 A. Domokos / Nonlinear Analysis 126 (2015) 131–142

where

cπ,λjk = dπ

Gf(x) πλjk(x) dµ(x).

Proof. Schur’s orthogonality relation (2.3) from Theorem 2.2 and the orthogonal decomposition (2.6) implythat

Eπ =λ∈ΛπEπλ ,

which combined with (2.4) from the Peter–Weyl theorem gives (2.7). The subelliptic Fourier series (2.8) isa direct consequence of (2.7).

As an immediate application of the two versions of the Peter–Weyl theorem, consider the elliptic heatequation on G,

∂f

∂t(t, x) = ∆f(t, x), f(0, x) = g(x) ∈ L2(G),

and the subelliptic heat equation on G,∂f

∂t(t, x) = ∆X f(t, x), f(0, x) = g(x) ∈ L2(G).

Theorems 2.3 and 2.4 give the following corollary.

Corollary 2.3. (a) The solution of the elliptic heat equation has the series form

f(t, x) =

[π]∈Gdπj,k=1

cπjk e−(∥wπ∥2+2⟨wπ,δ⟩)t πjk(x),

where

cπjk = dπ

Gg(x)πjk(x) dµ(x).

(b) The solution of the subelliptic heat equation has the series form

f(t, x) =

[π]∈Gλ∈Λπ

dπλj,k=1

cπ,λjk e−(∥wπ∥2+2⟨wπ,δ⟩−∥λ∥2)t πλjk(x),

where

cπ,λjk = dπ

Gg(x)πλjk(x) dµ(x).

In the following we propose to obtain forms of the Peter–Weyl theorems which are less dependent of thechoice of the basis of V π. For this, we denote the group convolution by

f1 ∗ f2(y) =

Gf1(x) f2(x−1y) dµ(x),

and the character of the representation π by

χπ(x) = traceπ(x).

A. Domokos / Nonlinear Analysis 126 (2015) 131–142 139

Theorem 2.5 (The Plancherel Theorem [11]). For any f ∈ L2(G) we have

f(x) =

[π]∈G

dπ f ∗ χπ(x), (2.9)

and

∥f∥L2(G) =

[π]∈G

(dπ)2 ∥f ∗ χπ∥2L2(G), (2.10)

where dπf ∗ χπ is the orthogonal projection of f onto Eπ.

A compact group is unimodular, so we can use the following result:

Proposition 2.3 ([15]). If f, ϕ, ψ ∈ L2(G) then

⟨f ∗ ϕ, ψ⟩L2(G) = ⟨f, ψ ∗ ϕ⟩L2(G) = ⟨ϕ, f ∗ ψ⟩L2(G), (2.11)

where f(x) = f(x−1).

To simplify our notations, in the following we drop the superscript λ and denote by eπj an orthonormalbasis of V π which corresponds to the orthogonal direct space decomposition (2.6) and hence it is the unionof orthonormal bases of all V πλ , λ ∈ Λπ. Let us define the subelliptic characters as

χπλ =eπj∈V πλ

πjj . (2.12)

Therefore,

χπ =λ∈Λπ

χπλ, (2.13)

and we can state the subelliptic form of Theorem 2.5.

Theorem 2.6 (The Subelliptic Plancherel Theorem). For any f ∈ L2(G) we have

f(x) =

[π]∈G

λ∈Λπ

dπ f ∗ χπλ(x), (2.14)

and

∥f∥L2(G) =

[π]∈G

λ∈Λπ

(dπ)2 ∥f ∗ χπλ∥2L2(G), (2.15)

where dπf ∗ χπλ is the orthogonal projection of f onto Eπλ .

Proof. First, (2.14) follows from (2.9) and (2.13).

Second, let us observe that by Theorem 2.1 and Proposition 2.3, if j = l then

πjk ∗πll(y) =

Gπjk(x) πll(x−1y) dµ(x) =

Gπjk(x) πll(y−1x) dµ(x)

=

G⟨π(x)eπj , eπk ⟩ ⟨π(x)eπl , π(y)eπl ⟩ dµ(x) = 0.

Hence,

⟨f ∗ πll, f ∗ πjj⟩L2(G) = ⟨πll, f ∗ f ∗ πjj⟩L2(G) = ⟨πll ∗ πjj , f ∗ f⟩L2(G) = 0, (2.16)

140 A. Domokos / Nonlinear Analysis 126 (2015) 131–142

and

⟨f ∗ πll, πjk⟩L2(G) = ⟨f, πjk ∗πll⟩L2(G) = 0. (2.17)

By (2.16), if λ1 = λ2, then f ∗ χπλ1 is orthogonal to f ∗ χπλ2 , which gives (2.15). Moreover, by (2.17), ifπjj ∈ Eπλ , then f ∗ πjj ∈ Eπλ and therefore f ∗ χπλ is the orthogonal projection of f onto Eπλ .

3. The case of SU(2)

Let us explore how the subelliptic versions of the Peter–Weyl and Plancherel theorems look in the caseof the special unitary group

SU(2) =xa,b =

a b

−b a

: a, b ∈ C, |a|2 + |b|2 = 1

.

Its Lie algebra is the three dimensional real vector space

su(2) =X =

ix1 x2 + ix3

−x2 + ix3 −ix1

: x1, x2, x3 ∈ R

.

The Killing form of su(2) is

K(X,Y ) = 4 trace (XY ),

while the inner product (1.1) is defined as

⟨X,Y ⟩ = −12 trace (XY ).

The Cartan subalgebra Γ is spanned by

T =i 00 −i

,

and the basis of the orthogonal complement of Γ is

X1 =

0 1−1 0

, X2 =

0 i

i 0

.

For integers m ≥ 0 consider the vector space of homogeneous polynomials of degree m,

Pm =P (z, v) =

mj=0

cjzjvm−j , c0, . . . , cm ∈ C

,

endowed with the inner product

⟨P,Q⟩ =S3P (z, v)Q(z, v) dσ(z, v),

where dσ is the normalized surface measure on S3. An orthonormal basis of Pm is formed by

Pmj (z, v) = αmj zjvm−j , 0 ≤ j ≤ m,

where

αmj =

m!

j!(m− j)! .

A. Domokos / Nonlinear Analysis 126 (2015) 131–142 141

A unitary, irreducible representation of SU(2) onto Pm can be defined as πm : SU(2)→ Aut(Pm),

πm(xa,b)P (z, v) = P ((z, v) · xa,b) = P (az − bv, bz + av).

It is a well-known result [10,11] that SU(2) = [πm] : m = 0, 1, 2, . . ., so we can start computing theeigenvalues and eigenvectors of the Casimir operators and Laplacians.

dπm(T )Pmj (z, v) = d

dt

t=0

πm(etT )Pmj (z, v)

= d

dt

t=0

Pmj

(z, v) ·

eit 00 e−it

= i(2j −m)Pmj (z, v).

Therefore, the weights of the representation dπm are λmj = (2j − m)T and the orthogonal direct spacedecomposition (2.6) is formed by the one dimensional subspaces

Vλmj

= spanPmj .

Continuing the calculations we get

dπm(X1)Pmj (z, v) = d

dt

t=0

πm(etX1)Pmj (z, v)

= d

dt

t=0

Pmj

(z, v) ·

cos t sin t− sin t cos t

= αmj

−jzj−1vm−j+1 + (m− j)zj+1vm−j−1 ,

and hence

dπm(X1)2Pmj (z, v) = αmj (j(j − 1)zj−2vm−j+2 − j(m− j + 1)zjvm−j

− (m− j)(j + 1)zjvm−j + (m− j)(m− j − 1)zj+2vm−j−2).

In a similar way,

dπm(X2)Pmj (z, v) = d

dt

t=0

πm(etX2)Pmj (z, v)

= d

dt

t=0

Pmj

(z, v) ·

cos t i sin ti sin t cos t

= αmj

ijzj−1vm−j+1 + i(m− j)zj+1vm−j−1 ,

and

dπm(X2)2Pmj (z, v) = αmj (−j(j − 1)zj−2vm−j+2 − j(m− j + 1)zjvm−j

− (m− j)(j + 1)zjvm−j − (m− j)(m− j − 1)zj+2vm−j−2).

In conclusion,

ΩmX Pmj = −(4mj − 4j2 + 2m)Pmj , (3.1)

and

ΩmPmj = −(m2 + 2m)Pmj . (3.2)

142 A. Domokos / Nonlinear Analysis 126 (2015) 131–142

The entry functions of the representation πm are

πmjk(xa,b) = ⟨πm(xa,b)Pmj , Pmk ⟩

=S3

(az − bv)j (bz − av)m−j zk vm−k dσ(z, v). (3.3)

By Proposition 2.2

∆Xπmjk = −(4mj − 4j2 + 2m)πmjk, (3.4)

and

∆πmjk = −(m2 + 2m)πmjk. (3.5)

Hence, for all f ∈ L2(SU(2)) we have

f (xa,b) =∞m=0

mj,k=0

cmjk πmjk (xa,b) , (3.6)

which corresponds to both the elliptic and subelliptic Peter–Weyl theorems and implicitly to the spectraldecomposition of both the elliptic and subelliptic Laplacian. Therefore, the solution of the elliptic heatequation in SU(2) has the form

f(t, xa,b) =∞m=0

mj,k=0

cmjke−(m2+2m)tπmjk(xa,b),

while the solution of the subelliptic heat equation looks like

f(t, xa,b) =∞m=0

mj,k=0

cmjke−(4mj−4j2+2m)tπmjk(xa,b).

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