The Dixmier trace and the Wodzicki residue for global pseudo-differential operators on compact manifolds

In this note, we announce the results of our investigation on the Dixmier trace and the Wodzicki residue for pseudo-differential operators on compact manifolds. We give formulae for the Dixmier trace and the non-commutative residue (also called Wodzicki’s residue) of invariant pseudo-differential operators on compact manifolds with or without boundary. For every closed manifold, the notion of global symbol for invariant pseudo-differential operators will be based on the Fourier analysis associated to every elliptic and positive operator (developed by M. Ruzhansky, V. Turunen and J. Delgado). In particular, for each compact Lie group we will use its representation theory. For the analysis of operators on compact manifolds with boundary, we will use the non-harmonic analysis associated with boundary valued problems (developed by M. Ruzhansky, N. Tokmagambetov, and J. Delgado).


Introduction
In this work, we classify the Dixmier traceability of some invariant operators on compact manifolds and we apply these results in order to study the Wodzicki residue of pseudodifferential operators. This work is a subsequent analysis to the program of M. Ruzhansky and J. Delgado on the classification of global operators in ideals contained in the algebra of bounded operators on Lebesgue spaces (see [16,17,18,19,20,21]). The topic has been of intense research in the last years not only for operators on compact manifolds, but also on non-compact manifolds and other spaces without a differentiable structure (see e.g. [1,3,4,5,6,8,9,10,11,14,15,27,24]). The novelty of the announced results is that we use the matrix-valued global quantisation developed by Ruzhansky, Turunen, Tokmagambetov and Delgado, instead of the classical results on the subject based on the classical notion of the local symbol via localizations (see, e.g. Hörmander [26]), as in [13,23,25,33,34] and references therein. We summarise our investigation in the following results: We provide necessary and sufficient conditions in order that global invariant pseudodifferential operators on a manifold with or without boundary belong to the Dixmier class L (1,∞) (L 2 ).
We provide sufficient and necessary conditions in order to obtain Dixmier traceability for a type of pseudo-differential operators with global symbols in the Hörmander classes on compact Lie groups, and we express our results by using the representation theory of these groups. Also, we use Connes' trace theorem in order to get formulae for the noncommutative residue of classical pseudo-differential operators on compact Lie groups.
For a smooth compact manifold with or without boundary, we find criteria in terms of the global symbols in order that the corresponding operators belong to the Marcinkiewicz ideal L (p,∞) (L 2 ), 1 < p < ∞, as well as to the Dixmier ideal Our main result will be presented in Section 3. In Section 2 we provide some basics on the theory of pseudo-differential operators, the Wodzicki residue and the Dixmier trace. Finally, in Section 4 we provide some examples.

Preliminaries
In order to present our main results we recall some definitions (see Connes [13]  where 1 < p < ∞. On the class L (p,∞) (L 2 ) the usual norm is given by The main objects here are pseudo-differential operators, which we introduce as follows ( [16,26,31]). A pseudo-differential operator A is an operator defined by the integral representation where the function σ A (x, ξ) -called the symbol of A-satisfies estimates of the type (see Hörmander [26]) Heref denotes the euclidean Fourier transform of the function f and m is the order of the pseudo-differential operator A. If we consider a compact manifold without boundary M of dimension κ, a pseudo-differential operator A on M can be defined by using the notion of local symbol; this means that for any local chart U , the operator A has the form A pseudo-differential operator A is called classical, if σ A admits an asymptotic expansion σ A (x, ξ) ∼ in ξ of order m − j for ξ = 0. The set of classical pseudo-differential operators of order m is denoted by Ψ m cl (M ). For A ∈ Ψ m cl (M ), and for x ∈ M , defines a local density which can be glued over M . In this case, the non-commutative residue of A is defined by the expression The following fact is due to Connes, [13]: if A ∈ Ψ −κ cl (M ) is a positive operator, then Tr ω (A) = res(A).
In the case of a manifold M with boundary, the typical algebra of pseudo-differential operators is known as the Boutet de Monvel algebra (BdM). This algebra is formed by matrix valued pseudo-differential operators associated with boundary value problems (see [2]): The noncommutative residue and the Dixmier trace for an operator P in BdM's algebra have been studied e.g. in Fedosov [23] and Nest and Schrohe [28].
In order to study the Dixmier trace and the non-commutative residue, we will use the notion of global pseudo-differential operator. If we consider a closed manifold M (i.e., a compact manifold without boundary) and H = L 2 (M ), there exists a global Fourier analysis associated to every positive elliptic pseudo-differential operator E on M which gives for certain pseudo-differential operators A, -called E-invariants-, a discrete Fourier representation of the form where e l (x) := (e m l ) 1≤m≤d l and {e m l : l ∈ N, 1 ≤ m ≤ d l } is a basis of L 2 (M ) consisting of eigenfunctions λ l , l ∈ N 0 , of E. The function σ A,E is called the matrix valued symbol of A with respect to E. The quantisation procedure associating to every operator A acting in C ∞ (M ) the matrix valued symbol σ A,E was introduced by J. Delgado and M. Ruzhansky, [20]. Also, If M = G is a compact Lie group and E = −L G is the positive Laplace-Beltrami operator on G, Ruzhansky and Turunen in [31] give a discrete representation to every pseudo-differential operator A acting on C ∞ (G) in terms of the representation theory of the group G, in the following way: Here G denotes the unitary dual of G. Ruzhansky-Turunen's calculus that gives a characterisation for the Hörmander classes Ψ m ρ,δ (G) on a compact Lie group G by using global symbols. In fact, A ∈ Ψ m ρ,δ (G), if and only if, its global symbol σ A (x, ξ) as in (5) Let us observe that if M = G is a compact Lie group and the operator E ∈ Ψ ν +e (G) is not invariant (this means that the global symbol σ E (x, ξ) depends of both variables, the spatial variable x ∈ G, and the Fourier variables [ξ] ∈ G), then in local coordinates, E-invariants operators A are E-multipliers, but they are not Fourier multipliers because in local coordinates their symbols are depending on the spatial variables.
In the formulation of our result for compact manifolds with boundary, we use the global quantization of pseudo-differential operators on compact manifolds with boundary due to J. Delgado, M. Ruzhansky and N. Tokmagambetov [30] which we briefly describe as follows. Let M be a compact manifold with boundary ∂M and let L be a pseudodifferential operator on M satisfying some boundary conditions on ∂M. We assume that L has discrete spectrum {λ ξ | ξ ∈ I}. If σ A,L : M × I → C is a suitable function, the operator A associated with σ A can be written as where f := F L (f ) denotes the L-Fourier transform of f introduced in [30]. L-Fourier multipliers are bounded linear operators A : , and for some function σ A,L : I → C depending only on the Fourier variables ξ ∈ I. In this case, σ A,L (ξ) is called the L-symbol of A.

The Dixmier Trace and the non-commutative residue for global pseudo-differential operators
In this section, we present our main results for global operators on compact manifolds. We start with the case of compact manifolds without boundary. Here, κ denotes the dimension of a fixed closed manifold.
where σ A,E is as in (4). Moreover, if A is positive, τ (A) = Tr w (A).
Let M = G be a compact Lie group and let G be the unitary dual of G. If we denote by σ A (x, ξ) the matrix valued symbol associated to A, then under the condition the operator A is Dixmier measurable.
If A ∈ Ψ −κ cl (G) is a classical and positive pseudo-differential operator and its symbol admits an asymptotic expansion in homogeneous components of the form: then The expression (11) in the preceding theorem gives a formula for the noncommutative residue of classical operators in terms of irreducible representations on G. We observe that our approach provide formulae for the noncommutative residue of operators on the torus, of a different way in relation with the recent work of Pietsch [29].
Now we present our result concerning to global operators on a manifold with boundary. With the notations above our result on the Dixmier traceability of operators on manifolds with boundary can be enunciated of the following way. We write I = {ξ l : l ∈ N 0 }. A is Dixmier traceable if, and only if, In this case, for A a positive operator, τ ′ (A) = Tr ω (A).
Moreover, if L is an operator of order m satisfying the Weyl eigenvalue counting formula, that is, then A is Dixmier traceable if, only if (cf. (12)), In this case, for A a positive operator, τ ′ (A) = Tr ω (A).
[Revista Integración, temas de matemáticas A ∈ L (p,∞) (L 2 (M )) if, and only if, where 1 < p < ∞. In this case, On the other hand, let A = [P + , T ] t be a column elliptic operator in the Boutet de Monvel algebra of order n. We assume the existence of L satisfying as in Ruzhansky-Tokmagambetov's calculus. Any parametrix R of P T is Dixmier traceable and Tr ω (R) = res (R) = lim N →∞ 1 ln N l≤N |(σ P,L (ξ l )) −1 |, here (σ P,L (ξ l )) l∈N0 denotes the global symbol of the operator P with respect to L. This residue's formula is the same for all parametrix of A, R and independent of the boundary condition and independent of the average function ω. If in addition L satisfies the Weyl Eigenvalue Counting Formula, we also have The proofs of these theorems and the corresponding analysis developed in this global setting can be found in Cardona and Del Corral [12]. We refer the reader to Cardona, Del Corral and Kumar [7] where the Dixmier trace of discrete pseudo-differential operators has been investigated.

Examples
In for all 0 < s ≤ s 1 < s ′ < ∞, then with the notations above the operator A := (I − L) − s 1 ν is Dixmier traceable on L 2 (M ). For the proof, we combine that the global symbol of A is given by As consequence of (18) we have that ξ l −s1 = O(l −1 ), l → ∞, and Hence, by Theorem 3.2 we obtain Under the assumption that a + b + 1 0 q(x) dx = 1, we have the inverse L −1 exists and is bounded from L 2 [0, 1] to D(L). The operator L has a discrete spectrum which can be enumerated by λ j = 2πj − i ln(−a/b) + α j , for j ∈ Z, where the sequence α j satisfies that for any ǫ > 0, j∈Z |α j | 1+ǫ < ∞. If m j denotes the multiplicity of the eigenvalue λ j , then m j = 1 for sufficiently large |j|, and the system of eigenfunctions are given by Let j 0 ∈ N large enough so that m j = 1 for |j| ≥ j 0 . The global symbol σ L,L (x, j) of L with respect to L is given by for some j 0 ∈ N. It follows from the functional calculus that L −1 has a discrete spectrum given by λ −1 j with j ∈ Z. Since |λ j | = O(j) for j ∈ Z, we have |j|≤N |λ −1 j | = O ln(2N +1) . Therefore, L −1 is a pseudo-differential operator which lies in L (1,∞) (L 2 [0, 1]) with symbol σ L −1 ,L (ξ) = (σ L,L (ξ)) −1 , |ξ| ≥ j 0 , and Example 4.3. Let us assume that A is a left-invariant operator on SU(2) ∼ = S 3 . We recall that the unitary dual of SU(2) (see [31]) can be identified as There are explicit formulae for t l as function of the Euler angles in terms of the so-called Legendre-Jacobi polynomials, see [31]. In this case, if A is Dixmier traceable with symbol σ([t l ]) ≡ σ(l), then For example, if A = (I −L SU(2) ) − κ 2 , is the Bessel potential of order −κ = − dim SU(2) = −3, then A is Dixmier traceable, σ(l) = (1 + l(l + 1)) − 3 2 and A similar result can be obtained if we consider the operator (I − L sub ) − 3 2 , where L sub := D 2 1 + D 2 2 is the sub-Laplacian on SO(3); here D 1 and D 2 are the derivatives in the corresponding variables to the Euler angles for SO(3) (see [31,32]).
Example 4.4. Let A be a left-invariant pseudo-differential operator Dixmier traceable on SU(3). The Lie group SU(3) has dimension 8 and 3 positive square roots α, β and ρ with the property ρ = 1 2 (α + β + ρ). We define the weights With the notations above, the unitary dual of SU(3) can be identified with The Dixmier trace of A, once denoted its full symbol by σ(λ) ≡ σ(λ(a, b)), is given by the  λ(a, b))|].
Remark 4. 5. Similar examples to the considered above can be obtained if we take, for example, every n-torus T n ∼ = R n /Z n or the compact Lie group SO(3) ∼ = RP 3 , spheres S n and arbitrary real and complex projective spaces.
Example 4.6. Let us assume that G is a compact Lie group of dimension κ, and let us define A := a(x)(−L G ) − κ 2 . We consider a(x) positive and smooth in order that A will be a positive operator. The global symbol of A is given by and by using (11) we have that In particular, for G = T κ , where T ≡ [0, 1), Consequently, which is a well known fact (see e.g. Connes [13]).