Lp-Nuclearity, traces, and Grothendieck-Lidskii formula on compact Lie groups

Given a compact Lie group $G$, in this paper we give symbolic criteria for operators to be nuclear and r-nuclear on Lp-spaces, with applications to distribution of eigenvalues and trace formulae. Since criteria in terms of kernels are often not effective in view of Carleman's example, in this paper we adopt the symbolic point of view. The criteria here are given in terms of the concept of matrix symbols defined on the non-commutative analogue of the phase space $G\times\hat{G}$, where $\hat{G}$ is the unitary dual of $G$.


Introduction
Let G be a compact Lie group. In this paper we address the following problems: • to find criteria for operators to be nuclear on L p (G), for 1 ≤ p < ∞; • since in the Banach spaces, due to Grothendieck's work [Gro55], we know that in order to have the operator trace to agree with the spectral trace, the notion of nuclearity is not sufficient, to find criteria for the r-nuclearity (0 < r ≤ 1) and to apply this to derive information on the spectral behaviour and on the traces of operators on L p (G). Our analysis will be based on the global quantization recently developed in [RT10a] and [RT12] as a noncommutative analogue of the Kohn-Nirenberg quantization of operators on R n .
In general, for trace class operators in Hilbert spaces, the trace of an operator given by integration of its integral kernel over the diagonal is equal to the sum of its eigenvalues. However, this property fails in Banach spaces. The notion of rnuclear operators becomes useful, and Grothendieck [Gro55] proved that 2 3 -nuclear operators in this scale satisfy the Lidskii trace formula on L p -spaces. The question of finding good criteria for ensuring the r-nuclearity of operators arises but this has to be formulated in terms different from those on Hilbert spaces and has to take into account the impossibility of certain kernel formulations in view of Carleman's example [Car16] recalled below.
In order to get an efficient criterion for the r-nuclearity, the application of the notion of a matrix symbol of an operator on a compact Lie will be instrumental. We also give several further applications. A special feature of our criteria is that we do not assume any regularity condition on the symbols, which shows a certain advantage in comparison with the traditional Kohn-Nirenberg quantization in the manifold setting. Here we will completely drop regularity assumption on the symbol as a consequence of the technique of noncommutative quantization that we are using. While Grothendieck's result yields the same index 2/3 for all L p spaces, we relate the index r of the r-nuclear operators with the index p of L p -spaces in which the trace formula holds. Nuclearity criteria for operators on L 2 with smooth symbols in Hörmander classes have been analysed, see e.g. Shubin [Shu01,Section 27]. The problem of finding criteria for Schatten classes in terms of symbols with lower regularity has been of interest in the last years, see e.g. [Tof06,Tof08,BT10].
Symbolic criteria for the L p -boundedness of operators on compact Lie groups, the Mikhlin-Hörmander multiplier theorem and its extension to non-invariant operators for 1 < p < ∞, are presented in [RW13].
To formulate the notions more precisely, let E and F be two Banach spaces and let 0 < r ≤ 1. A linear operator T from E to F is called r-nuclear if there exist sequences (x ′ n ) in E ′ and (y n ) in F so that x, x ′ n y n and (1.2) ∞ n=1 x ′ n r E ′ y n r F < ∞.
The class of r-nuclear operators is usually endowed with the pseudo-norm where the infimum is taken over the representations (1.1) of T such that (1.2) holds. When r = 1 we obtain the ideal of nuclear operators and n 1 (·) is a norm. In this case the definition above agrees with the concept of trace class operators in the setting of Hilbert spaces (E = F = H).
Since we are also interested in the distribution of eigenvalues we shall consider the case E = F and the notion of the trace. In order to ensure the existence of a good definition of the trace on the ideal of nuclear operators N(E) one is led to consider the Banach spaces E enjoying the so-called approximation property (cf. [Pie87], [DF93]). It is well known that the spaces L p (Ω, M, µ) satisfy the approximation property for any measure µ and 1 ≤ p ≤ ∞ (cf. [Pie80,Lemma 19.3.5]). Thus, a Banach space E is said to have the approximation property if for every compact subset K of E and every ǫ > 0 there exists a finite rank bounded operator B on E such that we have On such spaces, if T : E → E is nuclear, the trace is well-defined by x ′ n ⊗ y n is a representation of T as in (1.1). It can be shown that this definition is independent of the choice of the representation.
In the setting of Hilbert spaces the class of r-nuclear operators agrees with the Schatten-von Neumann ideal of order r, a result due to R. Oloff (cf. [Olo72]). When r = 2 3 , Grothendieck proved (cf. [Gro55]) that the trace in Banach spaces agrees with the sum of all the eigenvalues with multiplicities counted. In Hilbert spaces this holds for nuclear (i.e. trace class) operators, the result which is known as the Lidskii formula (cf. [Lid59]). It has been proven by A. Pietsch [Pie84] that if r > 1 the class of operators having decomposition (1.1) and satisfying (1.2) is essentially reduced to the null operator. The question about the sharpness of the index r = 2 3 for trace formulae in the case of L p -spaces has been recently considered by Reinov and Laif [RL11]. Being in the class of r-nuclear operators can be used to deduce properties concerning the asymptotic behaviour of the corresponding operators. The statement relating Grothendieck's r-nuclearity result to the Lidskii formula in L p spaces is known as a Grothendieck-Lidskii formula (see e.g. [RL11]) and we give its variant on compact Lie groups in Corollary 4.3 for 0 < r ≤ 2/3 and in Corollary 4.5 for 0 < r ≤ 1 with 1 r = 1 + 1 2 − 1 p . The r-nuclear operators are sometimes known as p-nuclear operators, but here we will reserve the index p to indicate the L p -spaces. A description of the current state of the art of the general theory of p-nuclear operators has recently appeared in Hinrichs and Pietsch [HP10].
Among other things, in this paper we establish sufficient conditions on the matrixvalued symbol of an operator in order to ensure the r-nuclearity in L p -spaces. The nuclearity of pseudo-differential operators on the circle T 1 has been recently analysed in [DW] but the situation in the present paper is much more subtle because of the necessarily appearing multiplicities of the eigenvalues of the Laplacian on the noncommutative compact Lie groups; moreover, due to the commutativity of the torus, the symbol there is scalar and hence all of its "matrix"-norms are uniformly equivalent which is not the case if the group G is noncommutative.
We shall now briefly recall a classical result of Carleman ([Car16]) which will be helpful to clarify the significance of our symbolic criteria. In 1916 Torsten Carleman T f = f * κ acting on L 2 (T 1 ) one obtains that the sequence (c n ) n forms a complete system of eigenvalues of this operator corresponding to the complete orthonormal system φ n (x) = e 2πinx , T φ n = c n φ n . The system φ n is also complete for T * , T * φ n = c n φ n , the singular values of T are given by s n (T ) = |c n | and hence ∞ n=−∞ s n (T ) r = ∞, for r < 2. Hence, the operator T is not nuclear. Moreover, due to the aforementioned Oloff's result the operator T is not r-nuclear for 0 < r ≤ 1. However, the continuous integral kernel k(x, y) = κ(x − y) satisfies any kind of integral condition of the form |k(x, y)| s dxdy < ∞ due to the boundedness of k. This shows that it is impossible to formulate a sufficient condition of this type for the kernel ensuring nuclearity on the torus T 1 .
In this work we will establish conditions imposed on symbols instead of kernels ensuring the r-nuclearity of the corresponding operators. The criteria that we will obtain in the general case for the nuclearity from L p 1 (G) to L p 2 (G) will depend on whether p 1 ≤ 2 or p 1 ≥ 2. For p 1 = p 2 we will apply this to the question of the convergence of the series of eigenvalues of operators and to the validity of the Lidskii formula. We give examples applying our results to the heat kernel on general compact Lie groups (Subsection 4.1) as well as to the Laplacian and the sub-Laplacian on SU(2) ≃ S 3 and on SO(3) (Subsection 3.2).
In Section 2 we discuss and formulate the known criteria for nuclearity as well as make a short introduction to the global quantization on compact Lie groups. In Section 3 we move to the setting of L p -spaces and formulate our criteria for the r-nuclearity. There are different possibilities of how to impose conditions on the symbol. We will discuss both the cases of invariant and non-invariant operators, and give examples of our results on the tori, on the group SU(2) and on SO(3). In Section 4 we give applications to summability of eigenvalues, trace formulae and the Lidskii theorem. In particular, in Subsection 4.1 we give the example of the trace kernel and its trace.
The authors would like to thank Jens Wirth for discussions and remarks.

Preliminaries
In this section we recall some basic facts about the concepts of nuclear and rnuclear operators, and the notion of the trace on Banach spaces. In particular, we consider the trace of nuclear operators on L p (µ). The fact that these spaces satisfy the approximation property is a classical result (cf. [Gro55], [Pie80]). We refer the reader to [Pie80] and to [Pie87,Chapter 4.2] for the general theory of traces on operator ideals and the notation used in this section, see also [GGK00] for an exposition on the distribution of the eigenvalues. For the theory of pseudo-differential operators on compact Lie groups the we refer to [RT10a] and [RT12].
In the case of L p -spaces we first record the following characterisation of nuclear operators (cf. [Del10]). In the statement below we shall consider (Ω 1 , M 1 , µ 1 ) and (Ω 2 , M 2 , µ 2 ) to be two σ-finite measure spaces.
Theorem 2.1. Let 1 ≤ p 1 , p 2 < ∞ and let q 1 be such that 1 p 1 + 1 q 1 = 1. An operator T : L p 1 (µ 1 ) → L p 2 (µ 2 ) is nuclear if and only if there exist sequences (g n ) n in L p 2 (µ 2 ), and (h n ) n in L q 1 (µ 1 ) such that ∞ n=1 g n L p 2 h n L q 1 < ∞, and such that for all f ∈ Remark 2.2. An analogue of the characterisation above holds for r-nuclear operators replacing the terms g n L p 2 h n L q 1 by g n r L p 2 h n r L q 1 in the sum. A distribution of the eigenvalues for r-nuclear operators can be obtained from the next theorem relating the eigenvalues and the class of r-nuclear operators (cf. Theorem 2.3. Let E be a Banach space which has the approximation property. Let T be an r-nuclear operator from E into E for some 0 < r ≤ 1. Then where λ n (T ) denote the eigenvalues of T with multiplicities counted.
Remark 2.4. (i) Note that from 1 s = 1 r − 1 2 we obtain that s = 2r 2−r for 0 < r ≤ 1. In particular, the function s(r) = 2r 2−r has the range (0, 2]. It is clear that if s > 2 the series on the left in Theorem 2.3 also converges but the interesting situation is to find smaller values of s ensuring such convergence. (ii) Theorem 2.3 was established by Grothendieck [Gro55], and later extended by e.g. König ([Kön78, page 107]) to the scale of Lorentz sequences spaces; see also [Pie87, Theorem 3.8.6].
(iii) If r = 1 we get s = 2, a classical result by Grothendieck (cf. [Gro55]) establishing the square summability of eigenvalues for nuclear operators. It is also known by Grothendieck that s = 2 is the best possible exponent in this case. Theorem 2.3 will be applied jointly with our sufficient conditions for r-nuclearity, to obtain estimates on the asymptotic behaviour of the eigenvalues. From this point of view, the main goal of this paper becomes to find suitable criteria for ensuring the r-nuclearity of an operator.
Given a compact Lie group G, in this work we consider Ω 1 = Ω 2 = G and M = M 1 = M 2 , the Borel σ-algebra associated to the topology of the smooth manifold G, with µ = µ 1 = µ 2 the normalised Haar measure of G. Let G denote the set of equivalence classes of continuous irreducible unitary representations of G. Since G is compact, the set G is discrete. For [ξ] ∈ G, by choosing a basis in the representation space of ξ, we can view ξ as a matrix-valued function ξ : G → C d ξ ×d ξ , where d ξ is the dimension of the representation space of ξ. By the Peter-Weyl theorem the collection is an orthonormal basis of L 2 (G). If f ∈ L 1 (G) we define its global Fourier transform at ξ by where dx is the normalised Haar measure on G. Thus, if ξ is a matrix representation, we have f (ξ) ∈ C d ξ ×d ξ . The Fourier inversion formula is a consequence of the Peter-Weyl theorem, so that we have Given a sequence of matrices a(ξ) ∈ C d ξ ×d ξ , we can define where the series can be interpreted distributionally or absolutely depending on the growth of (the Hilbert-Schmidt norms of) a(ξ). For a further discussion we refer the reader to [RT10a].
For each [ξ] ∈ G, the matrix elements of ξ are the eigenfunctions for the Laplacian L G (or the Casimir element of the universal enveloping algebra), with the same eigenvalue which we denote by −λ 2 [ξ] , so that we have The weight for measuring the decay or growth of Fourier coefficients in this setting is 1 2 , the eigenvalues of the elliptic first-order pseudo-differential operator (I − L G ) 1 2 . The Parseval identity takes the form which gives the norm on ℓ 2 ( G).

For a linear continuous operator
in the sense of distributions, and the sum is independent of the choice of a representation ξ from each equivalence class [ξ] ∈ G. If A is a linear continuous operator from C ∞ (G) to C ∞ (G), the series (2.7) is absolutely convergent and can be interpreted in the pointwise sense. We will also write A = Op(σ A ) for the operator A given by the formula (2.7). The symbol σ A can be interpreted as a matrix-valued function on G× G. We refer to [RT10a,RT12] for the consistent development of this quantization and the corresponding symbolic calculus. If the operator A is left-invariant then its symbol σ A does not depend on x. We often call such operators simply invariant.
We now record simple inequalities on the norms of the representation coefficients which will be essential for the analysis of r− nuclearity: Proof. If q = ∞, for any y ∈ G we have |ξ(y) ij | ≤ ξ(y) op = 1 by the unitarity of representations in G. If 2 ≤ q < ∞ we apply the inequality ξ , and that we have just showed that ξ ij L ∞ ≤ 1, we obtain Finally, for 1 ≤ q ≤ 2, using Hölder's inequality, we get where we have used that the Haar measure on G is normalised.
Our criteria will be formulated in terms of norms of the matrix-valued symbols. In order to justify the appearence of them, we recall that if A ∈ Ψ m (G) is a pseudodifferential operators in Hörmander's class Ψ m (G), i.e. if all of its localisations to R n are pseudo-differential operators with symbols in the class S m 1,0 (R n ), then the matrix-symbol of A satisfies Here · op denotes the operator norm of the matrix multiplication by the matrix σ A (x, ξ). For this fact, see e.g. [RT10a, Lemma 10.9.1] or [RT12], and for the complete characterisation of Hörmander classes Ψ m (G) in terms of matrix-valued symbols see also [RTW10]. In particular, this motivates the employ of the operator norms of the matrix-valued symbols. However, since σ A is in general a matrix, other matrix norms become useful as well.

r-Nuclearity on L p (G) and examples
In this and next sections we analyse the r-nuclearity and trace formulae in L pspaces. We recall that the case r = 1 correspond to the class of nuclear operators. One of the features of the obtained criteria is that they require the integrability (in some L p -space) of symbols σ A (x, ξ) with respect to x but do not assume any regularity of the symbol.
We start by proving the following sufficient condition for r-nuclearity of operators on L 2 (G) with symbols depending only on ξ. We note that the property that the symbol depends only on ξ means that the operator is left-invariant, that is, it commutes with the left translations on the group G.
Theorem 3.1. Let G be a compact Lie group. Let A : L 2 (G) → L 2 (G) be a linear continuous operator with matrix-valued symbol σ A (ξ) depending only on ξ. Then A is r-nuclear provided that its symbol σ A satisfies Here Proof. Let us suppose that the symbol σ A satisfies (3.1). We note that the kernel of the operator A is given by and we will show that it is well-defined and has the tensor product form of Theorem 2.1 that is required for the nuclearity. To abbreviate the notation, we will write σ(ξ) for σ A (ξ). We begin by writing and we set For g ξ,ij (x) we have completing the proof.
Remark 3.2. We point out that one can prove that the condition (3.1) ensuring the r-nuclearity for left-invariant operators on L 2 (G) is also necessary. We recall that in the Hilbert space setting, the Schatten class of order r agrees with the class of r-nuclear operators whenever 0 < r ≤ 1 by Oloff's result [Olo72]. For the details of Schatten classes of invariant operators on compact Lie groups we refer the reader to the recent work [DR13].
We will extend Theorem 3.1 to the setting of L p (G) spaces. We shall require the following notation: ℓ ∞ denotes the L ∞ -norm on C d ξ and · op(ℓ ∞ ,ℓ ∞ ) denotes the operator norm with respect to ℓ ∞ on C d ξ . More precisely, for each d ∈ N, let B ∈ C d×d and u ∈ C d . Denoting justifying the notation · op(ℓ ∞ ,ℓ ∞ ) , and the appearance of this norm. The transpose of the matrix M will be denoted by M t . We first deal with left-invariant operators.

<∞.
We now suppose p 2 > 2. Now, if p 2 > 2 we first observe that Hence and taking into account that |ξ(x) ik | ≤ 1, we get Then using (3.2) and (3.3) we obtain completing the proof.
In the particular case of diagonal symbols only depending on ξ we can improve the sufficient condition in the above theorem. An example of such behaviour is the sub-Laplacian on G that always has a diagonal symbol in an appropriately chosen basis in the representation spaces. Moreover, symbols of left-invariant self-adjoint operators can be chosen to be diagonal by choosing a particular representative from each equivalence class [ξ] ∈ G. We formulate a general result now, and will give its application to the sub-Laplacian in Subsection 3.2.
We will sometimes give examples of our results on the torus, so we summarise its notation: Remark 3.5. If G = T n = R n /Z n , we have T n ≃ Z n , and the collection {ξ k (x) = e 2πix·k } k∈Z n is the orthonormal basis of L 2 (T n ), and all d ξ k = 1. If an operator A is invariant on T n , its symbol becomes σ A (ξ k ) = ξ k (x) * Aξ k (x) = Aξ k (0). In general, on the torus we will often simplify the notation by identifying T n with Z n , and thus writing ξ ∈ Z n instead of ξ k ∈ Z n . The toroidal quantization has been analysed extensively in [RT10b] and it is a special case of (2.7), where we have identified, as noted, T n with Z n .
As a consequence of Theorem 3.1 on the torus, we obtain: Corollary 3.6. Let 1 ≤ p 1 , p 2 < ∞. Let A : L p 1 (T n ) → L p 2 (T n ) be a linear continuous operator with symbol σ A (ξ) depending only on ξ. Then A is r-nuclear provided that its symbol σ A satisfies We shall now consider more general operators so that the symbols may depend also on x.
Theorem 3.7. Let G be compact Lie group and 0 < r ≤ 1. Let operator A have the matrix symbol σ A (x, ξ). Let 1 ≤ p 1 , p 2 < ∞ and let us denote p 1 = min{2, p 1 }. Suppose that the symbol σ A satisfies Then the extension A : Proof. The kernel of the operator A is given by and we will show that it is well-defined and has the tensor product form of Theorem 2.1 that is required for the nuclearity. As before, to abbreviate the notation, we will write σ(x, ξ) for σ A (x, ξ). We begin by writing (ξ(x)σ(x, ξ)) ij ξ(y) ij , and we set A similar argument like in (3.3) shows that Let q 1 be such that 1 p 1 + 1 q 1 = 1. Now, if we denote q 1 = max{2, q 1 }, we have 1 p 1 + 1 q 1 = 1. According to (2.8), we have completing the proof.
As a consequence of Theorem 3.7, recalling the notation on the torus in Remark 3.5, for the torus group G = T n , we have: Corollary 3.8. Let 1 ≤ p 1 , p 2 < ∞ and let A : Remark 3.9. (i) If G = T n = R n /Z n , an invariant operator A is a Fourier multiplier, Af (k) = a(k) f (k) with symbol σ A (ξ k ) = a(k), see Remark 3.5. Theorem 3.7 implies that if k∈Z n |a(k)| r < ∞, then the operator T is r-nuclear from L p 1 (T n ) to L p 2 (T n ) for all 1 ≤ p 1 , p 2 < ∞.
(iii) If p 1 = p 2 = 2 and A is a left-invariant operator on a compact Lie group G, it follows from Theorem 3.1 that if [ξ]∈ G d ξ σ A (ξ) S 1 < ∞, then A is a trace class operator on L 2 (G).
(iv) We note that the condition of Corollary 3.8 required the integrability of the symbol with respect to x and does not require any regularity.
In order to deduce some interesting consequences we will apply the following lemma proved in [DR12]: Lemma 3.10. Let G be a compact Lie group. Then we have This yields the following corollary, and in Remark 3.14 we note that the following orders are in general sharp.
Corollary 3.11. Let G be a compact Lie group of dimension n and let 0 < r ≤ 1. Let 1 ≤ p 1 , p 2 < ∞ and let us denote p 1 = min{2, p 1 }. Assume that The result now follows from Lemma 3.10 and Theorem 3.7.
As consequence of Theorem 3.4 and Lemma 3.10 we have: Corollary 3.12. Let G be a compact Lie group, 1 ≤ p 1 , p 2 < ∞ and 0 < r ≤ 1. Let A : L p 1 (G) → L p 2 (G) be a linear continuous operator with matrix-valued diagonal symbol σ A (ξ) depending only on ξ. If where p 1 = min{2, p 1 }, p 2 = max{2, p 2 }. Then the operator A : The result now follows from Lemma 3.10 and Theorem 3.4.
3.1. Example on the torus. We observe that on the torus T n criteria obtained in the above statements are in general sharp. In general, we recall that the relation of our setting to the special case of the torus was outlined in Remark 3.5, with examples given already in Corollary 3.6 and in Corollary 3.8.
To see the sharpness, we establish the following simple characterisation of the nuclearity for Bessel potentials on L 2 (T n ).
Proposition 3.13. Let ∆ be the Laplacian on the torus T n and let 0 < r ≤ 1. Then (I − ∆) − α 2 is r-nuclear on L 2 (T n ) if and only if αr > n.
Proof. The symbol of the operator T = (I − ∆) − α 2 is positive, hence T being a multiplier operator, it is positive definite and |T | = √ T * T = T . Thus, the singular values of T agree with the values of its symbol ξ −α . Therefore, T ∈ S r (L 2 (T n ) if and only if αr > n. The result now follows from the identification of the Schatten class of order r and the class of r−nuclear operators [Olo72].
Remark 3.14. In the case of the torus T n we have d ξ = 1. From Proposition 3.13 it follows that the index n in the sufficient condition in Corollary 3.11 cannot be improved.
Using compactness of G, the following criterion can be practical: Corollary 3.16. Let 1 ≤ p 1 , p 2 < ∞, 0 < r ≤ 1 and let A : L p 1 (T n ) → L p 2 (T n ) be a linear continuous operator with symbol σ A (x, ξ) satisfying |σ A (x, ξ)| ≤ C ξ −s/r , for some s > n. Then the operator A : L p 1 (T n ) → L p 2 (T n ) is r-nuclear (for all p 1 , p 2 ).
3.2. Examples on SU(2) ≃ S 3 and on SO(3). Let us now show other examples of the above statements for some particular compact groups. We first consider the case of G = SU(2), the group of the unitary 2 × 2 matrices of determinant one. The same results as given below can be stated for the 3-sphere S 3 by using of the identification SU(2) ≃ S 3 , with the matrix multiplication in SU(2) corresponding to the quaternionic product on S 3 , with the corresponding identification of the symbolic calculus, see [RT10a, Section 12.5].
In particular we will apply the above corollary to the Laplacian and the sub-Laplacian.
We shall now consider the group SO(3) of the 3 × 3 real orthogonal matrices of determinant one. For the details of the representation theory and the global quantization of SO(3) we refer the reader to [RT10a,Chapter 12]. The dual in this case can be identified as G ≃ N 0 , so that The dimension of each t ℓ is d t ℓ = 2ℓ + 1. The Laplacian on SO(3) has eigenvalues λ 2 t ℓ = ℓ(ℓ + 1), so that we have t ℓ ≈ ℓ. By the same argument as above, Corollary 3.19 also holds for the Laplacian on SO(3).
Let us fix three invariant vector fields D 1 , D 2 , D 3 on SO(3) corresponding to the derivatives with respect to the Euler angles. We refer to [RT10a,Chapter 11] for the explicit formulae for these. However, for our purposes here we note that the sub-Laplacian L sub = D 2 1 + D 2 2 , with an appropriate choice of basis in the representation spaces, has the diagonal symbol given by (3.6) σ L sub (ℓ) mn = (m 2 − ℓ(ℓ + 1))δ mn , m, n ∈ Z, −ℓ ≤ m, n ≤ ℓ, where δ mn is the Kronecker delta. The operator L sub is a second order hypoelliptic operator and we can define the powers (I − L sub ) −α/2 . These are pseudo-differential operators with symbols σ (I−L sub ) −α/2 (ℓ) mn = (1 + ℓ(ℓ + 1) − m 2 ) −α/2 δ mn .

Trace formulae on L p (G) and distribution of eigenvalues
We now turn to some applications of the r-nuclearity on L p (G) spaces for the trace formulae, the Lidskii formula and the distribution of eigenvalues. In the special case 1 ≤ p 1 = p 2 = p < ∞, applying Theorem 2.3 and Theorem 3.7 we obtain: Corollary 4.1. Let G be compact Lie group and 0 < r ≤ 1. Let 1 ≤ p < ∞ and let us denote p = min{2, p}. Let σ A (x, ξ) be the matrix symbol of a bounded operator Then A : We now derive another consequence relating the trace formulae with matrix-valued symbols. As we have already explained in the introduction, every nuclear operator defined from a Banach space E into E admits a trace provided that E satisfies the approximation property, which is the case here dealing with L p spaces. In the next proposition we show that, when p = p 1 = p 2 the sufficient condition in Theorem 3.7 ensures the existence of a formula for the trace in terms of the matrix-valued symbol.
Theorem 4.2. Let G be a compact Lie group and 0 < r ≤ 1. Let 1 ≤ p < ∞ and p = min{2, p}. Let A : L p (G) → L p (G) be a linear continuous operator with matrix-valued symbol σ A (x, ξ) such that Then the operator A : L p (G) → L p (G) is r-nuclear and its trace is given by Proof. The r-nuclearity is a consequence of Theorem 3.7 and we adopt the notation of the proof of Theorem 3.7, and denote σ = σ A . Concerning the trace formula, in sake of simplicity we will just consider r = 1 the general case follows from inclusion. As we have seen in the proof of Theorem 3.7, the formula represents the kernel of A. Moreover, it is well defined on the diagonal: in fact for the terms of the decomposition of the kernel g ξ,ij (x) = d ξ (ξ(x)σ(x, ξ)) ij , h ξ,ij (y) = (ξ(y) * ) ji = ξ(y) ij , by Hölder's inequality we have on the diagonal Hence, since p = p 1 = p 2 we have Therefore, We have employed the tracial property Tr(AB) = Tr(BA) and the fact that ξ(x) is unitary for every x.
In relation with the Lidskii formula, from Theorem 4.2 and Grothendieck's theorem we have: Corollary 4.3. Let G be a compact Lie group and 0 < r ≤ 2 3 . Let 1 ≤ p < ∞ and p = min{2, p}. Let A : L p (G) → L p (G) be a linear continuous operator with matrix-valued symbol σ A (x, ξ) such that Then the operator A : L p (G) → L p (G) is r-nuclear and its trace satisifies with multiplicities taken into account.
Remark 4.4. We note that not for every kernel it is convenient to calculate the trace integrating along the diagonal due to the degeneracy of the measure on it. When the kernel is representable by an expansion of the kind appearing in Theorem 2.1 one is allowed to proceed in such a way. For a general kernel the integration along the diagonal should be calculated involving an averaging processes, see e.g. [Del10].
Very recently it has been proved (cf. [RL11]) that if 1 r = 1 + | 1 2 − 1 p |, the Lidskii formula holds for r-nuclear operators on L p (ν)-spaces. The importance of this result for us is that it allows to move r along the interval [ 2 3 , 1] keeping the validity of Lidskii's formula for suitable values of p. If r ∈ ( 2 3 , 1) there exist two corresponding values of p solving the equation 1 r = 1 + | 1 2 − 1 p | the first one with p < 2 and the other one with p > 2. As a consequence of this result and Theorem 4.2 we obtain an extension of Corollary 4.3 allowing now a larger range of r: Corollary 4.5. Let G be compact Lie group. Let 1 ≤ p < ∞ and let us denote p = min{2, p}. Let 0 < r ≤ 1 be such that 1 r = 1 + | 1 2 − 1 p |. If then A is r-nuclear on L p (G) and we have with multiplicities taken into account.
4.1. Heat kernels. We shall now establish some applications, in particular to the heat kernels on compact Lie groups. The heat kernel constructions, and the subsequent Poisson kernel constructions, are instrumental in the advances in the Littlewood-Paley theory on compact Lie groups, see e.g. [Ste70]. However, our approach is more straightforward, making use of the symbol of the heat kernel. Indeed, taking into account that σ e −tL G (x, ξ) = e −t|ξ| 2 I d ξ , where |ξ| 2 = λ 2 We can now derive the nuclearity of the heat kernel on L p -spaces.
Theorem 4.6. Let G be compact Lie group. Then the heat operator e −tL G : L p 1 (G) → L p 2 (G) is nuclear for every t > 0 and all 1 ≤ p 1 , p 2 < ∞. Moreover, if 0 < r ≤ 1, then e −tL G : L p (G) → L p (G) is r-nuclear for every t > 0 and 1 ≤ p < ∞. In particular, on each L p (G), due to the 1-nuclearity we have the trace formula Proof. The kernel of e −tL G is given by ξ(x) ij ξ(y) ij .
We set g ξ,ij (x) = d ξ e −tλ 2 [ξ] ξ(x) ij , h ξ,ij (y) = (ξ(y) * ) ji = ξ(y) ij . As before we shall consider q 1 such that 1 p 1 + 1 q 1 = 1 and we denote q 1 = max{2, q 1 }. Then by Lemma 2.5 we have On the other hand g ξ,ij L p 2 (G) = d ξ e −tλ 2 [ξ] ξ ij L p 2 (G) the last convergence following, for example, from any of the Weyl formulae, see, for example [DR12]. The r-nuclearity follows in a similar way. The trace formula follows immediately from Lemma 4.2 and fact that the Haar measure on G is normalised: As a consequence of this observation, taking into account the trace formula in [Del10] we obtain the following estimate for integral operates. The symbol will denote the averaging process for kernels described in [Del10]. Let µ be a Borel measure on a second countable topological space and let T : L p (µ) → L p (µ) be a 2 3 -nuclear operator with kernel K(x, y). If T possesses at least one eigenvalue and if |λ k (T )| ≤ M for all k, then | where N is the number of eigenvalues of T . The last inequality means that the better one can estimate the size of the trace the better lower bound one gets for the number of the eigenvalues.