Approximation property and nuclearity on mixed-norm $L^p$, modulation and Wiener amalgam spaces

In this paper we first prove the metric approximation property for weighted mixed-norm $L_w^{(p_1,\dots ,p_n)}$ spaces. Using Gabor frame representation this implies that the same property holds in weighted modulation and Wiener amalgam spaces. As a consequence, Grothendieck's theory becomes applicable, and we give criteria for nuclearity and $r$-nuclearity for operators acting on these space as well as derive the corresponding trace formulae. Finally, we apply the notion of nuclearity to functions of the harmonic oscillator on modulation spaces.


Introduction
The modulation spaces were introduced in 1983 by H. Feichtinger [Fei83] and have been intensively investigated in the last decades. We refer the reader to the survey [Fei06] by Feichtinger for a historical account of the development of such spaces and a good account of the literature. We also refer to Gröchenig's book [Grö01] for the basic definitions and properties of modulation spaces. Modulation spaces start finding numerous applications in various problems in linear and nonlinear partial differential equations, see [RSW12] for a recent survey.
The analysis of the Schatten properties of pseudo-differential operators acting on L 2 but with symbols of low regularity has been a subject of intensive recent research too, see e.g. Toft [Tof06,Tof08], Sobolev [Sob14], or the authors' papers [DR14b,DR14a]. In particular, Schatten properties of pseudo-differential operators with symbols in modulation spaces have been analysed and established by Toft [Tof04a,Tof04b], Gröchenig and Toft [GT14], see also [GT11].
One purpose of this paper is to analyse the analogous properties of operators but this time acting on modulation spaces. Since these are in general Banach spaces, the Schatten properties are replaced by the notion of nuclearity (or r-nuclearity) introduced by Grothendieck [Gro55]. In order for this theory to become effective we prove that the modulation spaces (and hence also Wiener amalgam spaces) have the so-called approximation property. This is done by proving the same property for weighted mixed-norm Lebesgue spaces and then using the Gabor frame description of modulation spaces reducing them to weighted mixed-norm sequence spaces. Consequently, we derive criteria for nuclearity and r-nuclearity of operators acting on modulation spaces with the subsequent trace formulae. The obtained results are applied to study functions of harmonic oscillator on modulation spaces and the corresponding trace formula of Lidskii type, relating the operator trace to the sums of eigenvalues for these operators.
We note that the approximation properties by themselves constitute fundamental 'Banach' properties of Banach spaces, see e.g. Figiel, Johnson and Pelczyński [FJP11] for a recent review of the subject, as well as Pietsch's book [Pie07,Section 5.7.4] for a survey of different approximation properties and relations among them as well as for the historical perspective. Recently, Alberti, Csörnyei, Pelczyński and Preiss [ACPP05] established the bounded approximation property (BAP) for functions of bounded variations, and Roginskaya and Wojciechowski [RW14] for Sobolev spaces W 1,1 . We refer to these papers for further references and motivations. In the present paper we prove the metric approximation property (which, in particular, implies the bounded approximation property because the control of the constant is explicit) for weighted mixed-norm Lebesgue spaces, modulation, and Wiener amalgam spaces, and apply this property further to derive spectral information about operators acting on these spaces.
To formulate the concepts more precisely, we now recall the notion of modulation spaces and that of nuclear operators on Banach spaces. For a suitable weight w on R 2d , 1 ≤ p, q < ∞ and a window g ∈ S(R d ) the modulation space denotes the short-time Fourier transform of f with respect to g at the point (x, ξ). The modulation space M p,q w (R d ) endowed with the above norm becomes a Banach space, independent of g = 0.
We now recall the required basic conditions on w for the development of the theory of modulation spaces M p,q w , and we refer the reader to the Chapter 11 of [Grö01] for a detailed exposition. A weight function is a non-negative, locally integrable function In particular the weights of polynomial type play an important role. They are of the form v s (x, ξ) = (1 + |x| 2 + |ξ| 2 ) s/2 .
(1.4) The v s -moderated weights (for some s) are called polynomially moderated.
On the other hand, the approximation property on a Banach space is crucial to define the concept of trace of nuclear operators and in particular for the study of trace formulae such as the Grothendieck-Lidskii formula. Let B be a Banach space, a linear operator T from B to B is called nuclear if there exist sequences (x ′ n ) in B ′ and (y n ) in B such that This definition agrees with the concept of trace class operator in the setting of Hilbert spaces. The set of nuclear operators from B into B forms the ideal of nuclear operators N (B) endowed with the norm It is natural to attempt to define the trace of T ∈ N (B) by x ′ n ⊗ y n is a representation of T . Grothendieck [Gro55] discovered that Tr(T ) is well defined for all T ∈ N (B) if and only if B has the aproximation property (cf. Pietsch [Pie87] or Defant and Floret [DF93]), i.e. if for every compact set K in B and for every ǫ > 0 there exists F ∈ F (B) such that where we have denoted by F (B) the space of all finite rank bounded linear operators on B. We denote by L(B) the C * -algebra of bounded linear operators on B. There are more related approximation properties, e.g. if in the definition above the operator F satisfies F ≤ 1 one says that B possesses the metric approximation property. This is closely related to the bounded approximation property, see e.g. Lindenstrauss and Tzafriri [LT77, Definition 1.e.11].
It is well known that the classical spaces C(X) where X is a compact topological space, as well as L p (µ) for 1 ≤ p < ∞ for any measure µ satisfy the metric approximation property (cf. Pietsch [Pie80]). In [Enf73] Enflo constructed a counterexample to the approximation property in Banach spaces. A more natural counterexample was then found by Szankowski [Sza81] who proved that B(H) does not have the approximation property.
An important feature on Banach spaces even endowed with the approximation property is that the Lidskii formula does not hold in general for nuclear operators, as it was proved by Lidskii [Lid59] in Hilbert spaces showing that the operator trace is equal to the sum of the eigenvalues of the operator counted with multiplicities. Thus, in the setting of Banach spaces, Grothendieck [Gro55] introduced a more restricted class of operators where Lidskii formula holds, this fact motivating the following definition.
Let B be a Banach space and 0 < r ≤ 1, a linear operator T from B into B is called r-nuclear if there exist sequences (x ′ n ) in B ′ and (y n ) in B so that x, x ′ n y n and ∞ n=1 x ′ n r We associate a quasi-norm n r (T ) by where the infimum is taken over the representations of T as in (1.6). When r = 1 the 1-nuclear operators agree with the nuclear operators, in that case this definition agrees with the concept of trace class operator in the setting of Hilbert spaces (B = H).
More generally, Oloff proved in [Olo72] that the class of r-nuclear operators coincides with the Schatten class S r (H) when B = H and 0 < r ≤ 1. Moreover, Oloff proved that where · Sr denotes the classical Schatten quasi-norms in terms of singular values.
In [Gro55] Grothendieck proved that if T is 2 3 -nuclear from B into B for a Banach space B endowed with the approximation property, then where λ j (j = 1, 2, . . . ) are the eigenvalues of T with multiplicities taken into account, and Tr(T ) is as in (1.5). Nowadays the formula (1.8) is refered to as Lidskii's formula, proved by V. Lidskii [Lid59] in the Hilbert space setting. Grothendieck also established its applications to the distribution of eigenvalues of operators in Banach spaces. We refer to [DR14b] for several conclusions in the setting of compact Lie groups concerning summability and distribution of eigenvalues of operators on L pspaces once we have information on their r-nuclearity. Kernel conditions on compact manifolds have been investigated in [DR14c], [DR14a]. For our purposes it is convenient to consider first mixed-norm spaces. In Section 2 we establish the metric approximation property for the weighted mixed-norm L (p 1 ,...,pn) w spaces, and in Section 3 for modulation spaces M p,q w and Wiener amalgam spaces W p,q w , also recalling the definition of the latter. In Section 4 we characterise r-nuclear operators acting between weighted mixed-norm spaces L P w . In Section 5 we apply it to the questions of r-nuclearity and trace formulae in modulation spaces and functions of the harmonic oscillator in that setting.

L
(p 1 ,...,pn) w has the metric approximation property In this section we prove that mixed-norm spaces L (p 1 ,...,pn) w and consequently the modulation spaces M p,q w satisfy the metric approximation property. Through the Fourier transform, also the Wiener amalgam spaces W p,q w will have the same property.
We start the analysis of the approximation property by recalling a basic lemma which simplifies the proof of the metric approximation property (cf. [Pie80], Lemma 10.2.2).
Lemma 2.1. A Banach space B satisfies the metric approximation property if, given x 1 , . . . , x m ∈ B and ǫ > 0 there exists an operator F ∈ F (B) such that F ≤ 1 and We shall now establish the metric approximation property for weighted mixednorm spaces. We first briefly recall its definition and we refer the reader to [BP61] for the basic properties of these spaces.
Let (Ω i , S i , µ i ), for i = 1, . . . , n, be given σ-finite measure spaces. We write x = (x 1 , . . . , x n ), and let P = (p 1 , . . . , p n ) a given n-tuple with 1 ≤ p i < ∞. We say that 1 ≤ P < ∞ if 1 ≤ p i < ∞ for all i = 1, . . . , n. Let w be a strictly positive measurable function. The norm · L P w of a measurable function f (x 1 , . . . , x n ) on the corresponding product measure space is defined by As should be observed, the order of integration on these spaces is crucial. L P w -spaces endowed with the · L P w -norm become Banach spaces and the dual ( In view of our application to the modulation spaces M p,q w we will consider in particular the case of the index of the form (P, Q) = (p 1 , . . . , p d , q 1 , . . . , q d ) where p i = p, q i = q and Ω i = R endowed with the Lebesgue measure. In this case the weight is taken in the form w = w(x, ξ) where x ∈ R d , ξ ∈ R d , with some special conditions on w that we briefly recall at the end of this section.
We first establish a useful lemma for the proof of the metric approximation property. We will require some notations. For n ∈ N and P = (p 1 , . . . , p n ) we will denote by ℓ P (I) the L P mixed-norm space corresponding to I n where I is a countable set of indices endowed with the counting measure. We note that such ℓ P -norm is given by Given a Banach space B and u ∈ B, z ∈ B ′ we will also denote by u, z B,B ′ , or simply by u, z , the valuation z(u).
Lemma 2.2. Let B be a Banach space and 1 ≤ Q < ∞. Let I be a countable set.
Then the operator T = i∈I u i ⊗ v i from B into B is well defined, bounded and satisfies We have applied the Hölder inequality in ℓ Q mixed-norm spaces (cf. [BP61]) for the second inequality. Hence T = lim N T N exists in L(B) and T L(B) ≤ 1.
We will apply the lemma above for the particular case of the single index Q = p n and the Banach space L P . In the rest of this section we will assume that the weights w satisfy the following condition for all x ∈ Ω: where w j is a weight on Ω j (i.e. a strictly positive locally integrable function). In particular, the condition holds for polynomially moderate weights on R n satisfying for a suitable n-tuple (β 1 , . . . , β n ), the condition Theorem 2.3. The weighted mixed-norm spaces L P w = L (p 1 ,...,pn) w with w satisfying (2.1) have the metric approximation property.
Proof. (Step 1) We will first prove the metric approximation property for the constant weight w = 1. Let f 1 , . . . , f N ∈ L (p 1 ,...,pn) and let ǫ > 0. By Lemma 2.1 it is enough to construct an operator L ∈ F (L P , L P ) such that L L(L P ) ≤ 1 and We consider first elementary functions of the form where the sets denotes the characteristic function of the set For the density of simple functions in L P see [BP61]. Since we are excluding the index p = ∞ in the multi-index P , the density always holds in our context.
We will denote We define In order to prove that L L(L P ) ≤ 1 we will apply Lemma 2.2 for B = L P , the families u(k), v(k) and Q = p n . The special role of the power p n will become clear later. Let f ∈ L P , g ∈ L P ′ be such that f L P , g L P ′ ≤ 1. Then we have to show that f, u (k) ℓ pn ≤ 1 and v (k) , g ℓ p ′ n ≤ 1.
In order to verify the corresponding property for f ∈ L P , it is enough to consider an elementary function f ∈ L P such that f L P ≤ 1. The general case follows then by approximation. By redefining partitions, we can assume that f can be written in the form We note that On the other hand, a straightforward but long calculation shows that Since f L P ≤ 1, we have shown that f, u (k) ℓ pn ≤ 1. The proof for v (k) , g ℓ p ′ n is similar and we omit it.

Now we obtain (2.3) in view of
For the third equality we have used the fact that the sets n ℓ=1 1 Ω ℓ k ℓ are disjoint. ( Step 2) We will now prove the metric approximation property for an elementary weight w (in this case we ask for weights to be non-negative and locally integrable). We can write such w in the form where the sets n j=1 Ω j k j are disjoint, µ j (Ω j k j ) < ∞ and γ k 1 ,...,kn > 0 for all (k).
Let f 1 , . . . , f N ∈ L (p 1 ,...,pn) w and let ǫ > 0. With a slight modification of (Step 1) we will find an operator L ∈ F (L P w , L P w ) such that L L(L P w ) ≤ 1 and f i − Lf i L P w ≤ ǫ, for i = 1, . . . , N. (2.5) We consider again elementary functions, by redefining partitions they can be written in the form and We define and set Again, in order to prove that L L(L P w ) ≤ 1 we will apply Lemma 2.2 for B = L P and the families u(k), v(k) and Q = p n . We note that w −1 , defined as 1/w on the support of w, is also an elementary weight. Let f ∈ L P w , g ∈ L P ′ w −1 be such that In order to verify the corresponding property for f ∈ L P w , it is enough to consider an elementary function f ∈ L P such that f L P w ≤ 1. By redefining partitions, we can assume that f can be written in the form We note that We also have Since f L P w ≤ 1, we have shown that f, u (k) ℓ pn ≤ 1. The proof for v (k) , g ℓ p ′ n is similar.
Therefore L L(L P w ) ≤ 1. The rest of the proof follows as in (Step 1). (Step 3) We now suppose that the weight w belongs to the mixed-norm space L P (µ) for some 1 ≤ P < ∞. (γ The desired operator L can be obtained by defining L := lim m L m in L(L P w ). We note that since L m L(L P wm ) ≤ 1 one gets L L(L P w ) ≤ 1. (Step 4) Now if a general weight w satisfies (2.1), we have the estimate w(x 1 , . . . , x n ) ≤ w 1 (x 1 ) · · · w n (x n ) for some positive functions w j . Let ψ j (x j ) be a positive function of a variable x j only such that 1 ψ j ∈ L p (µ j ) for some 1 ≤ p < ∞. We denote w j := w j ψ j . Then we observe that by writing w(x 1 , . . . , x n ) = w(x 1 , . . . , x n ) w 1 (x 1 ) −1 · · · w n (x n ) −1 w 1 (x 1 ) · · · w n (x n ), we obtain for 1 ≤ P = (p, . . . ,p) < ∞ and µ = w 1 (x 1 )µ 1 ⊗ · · · ⊗ w n (x n )µ n .
Since w ∈ L P (µ) for some 1 ≤ P < ∞, by (Step 3) applied to L P w ( µ), the approximation property follows for L P w (µ). This concludes the proof of the theorem.
3. M p,q w and W p,q w have the metric approximation property It is also important to consider the special case of discrete weighted mixed-norm spaces. Given α, β > 0, a strictly positive function w on the lattice αZ d × βZ d , we denote by ℓ p,q w (Z 2d ) the set of sequences a = (a kl ) k,l∈Z d for which the norm is finite. The main example arises from restrictions of weights on R 2d to a lattice αZ d × βZ d and will be crucial in the next Corollary 3.1.
Let us recall now the definition of the Wiener amalgam spaces W p,q w (R d ). There are several definitions possible for the spaces W p,q w , in particular involving the short-time Fourier transform similarly to the definition of the modulation spaces in (1.1). To make an analogy with modulation spaces, we can reformulate their definition (1.1) in terms of the mixed-normed Lebesgue spaces, by saying that (3.2) However, for our purposes the following description through the Fourier transform will be more practical. For a review of different definitions we refer to [RSTT11]. So, in what follows, we will always assume that the weights in modulation and Wiener amalgam spaces are submultiplicative and polynomially moderate 1 as in (1.2)-(1.4). Then, because of the identity the Wiener amalgam space W p,q w and the modulation spaces are related through the Fourier transform by the formula where w(x, ξ) = w 0 (ξ, −x). As a consequence of Theorem 2.3 we now obtain: Corollary 3.1. Let 1 ≤ p, q < ∞, and w a submiltiplicative polynomially moderate weight. Then M p,q w has the metric approximation property. Consequently, also the Wiener amalgam space W p,q w has the metric approximation property. Proof. We first observe that the polynomially moderate weights satisfy conditions (2.1) and (2.2) by choosing β j ≥ s. Also, we have the (topological) equivalence M p,q w ∼ = ℓ p,q w (Z 2d ) with equivalence of norms, by [Grö01, Theorems 12.2.3, 12.2.4], where w is the restriction of w to the lattice αZ d × βZ d , i.e. w(j, k) = w(αj, βk). The result now follows from Theorem 2.3 by taking Ω 1 , . . . , Ω 2d = Z, the weight ω, µ 1 = · · · = µ 2d = µ (the counting measure). This proves the metric approximation property for M p,q w since that property is preserved under isomorphism.
The metric approximation property for the Wiener amalgam spaces now follows from that in modulation spaces in view of the relation (3.3).
4. r-nuclearity on weighted mixed-norm spaces L P w Since the metric approximation property is now established for the spaces of our interest, it is now relevant to consider nuclear operators on weighted mixed-norm spaces L P w . In this section we will characterise nuclear operators on L P w and present some applications to the study of the harmonic oscillator on modulation spaces M p,q w . We first formulate a basic lemma for special measures and weights. We will consider 1 ≤ P, Q < ∞. The multi-index P will be associated to the measures µ i (i = 1, . . . , l) and Q will correspond to the measures ν j (j = 1, . . . , m). We will also denote µ := µ 1 ⊗· · ·⊗µ l and ν := ν 1 ⊗· · ·⊗ν m the corresponding product measures on the product For a weight w we will denote w P (Ω) := 1 Ω L P w (µ) . The additional property (2.1) will be only required for the formulation of trace relations.
Proof. Let k n (x, y) := n j=1 g j (x)h j (y)f (y). Applying the Hölder inequality we obtain ≤M < ∞ for all n.
Using Levi monotone convergence theorem the limit s(x, y) = lim n s n (x, y) is finite for a.e. (x, y). Moreover s ∈ L 1 (ν ⊗ µ), choosing f = 1 which belongs to L P w (µ) and from the fact that |k(x, y)| ≤ s(x, y) we deduce (a) and (b). Part (c) can be deduced using Lebesgue dominated convergence theorem applied to the sequence (k n ) dominated by s(x, y). For the part (d) we observe that letting k n (x, y) = n j=1 g j (x)h j (y)f (y), we have |k n (x, y)| ≤ s(x, y) for all n and every (x, y). From the fact that s ∈ L 1 (ν ⊗µ) we obtain that s(x, ·) ∈ L 1 (µ) for a.e x. Then (d) is obtained from Lebesgue dominated convergence theorem.
Remark 4.2. In the case of a single measure space (l = 1), the condition w P (Ω) < ∞ is equivalent to the fact that wµ is a finite measure. In particular, if w = 1 we have w P (Ω) < ∞ if and only if µ is a finite measure.
We establish below a characterisation of nuclear operators on the weighted mixednorm spaces L P w for weights satisfying the assumptions of Lemma 4.1.
. . , m) be measure spaces. Let w, w be weights on Ω, Ξ respectively satisfying conditions w P (Ω), w −1 Q ′ (Ξ) < ∞. Then T is r-nuclear operator from L P w (µ) into L Q w (ν) if and only if there exist a sequence (g n ) in L Q w (ν), and a sequence (h n ) in L P ′ w −1 (µ) such that Proof. It is enough to consider the case r = 1. The case 0 < r < 1 follows by inclusion. Let T be a nuclear operator from L P w (µ) into L Q w (ν). Then there exist where the sums converges in the L Q w (ν) norm. There exists (cf. [BP61], Theorem 1(a)) two sub-sequences ( g n ) and ( h n ) of (g n ) and (h n ) respectively such that Since the couple ( g n ), ( h n ) satisfies Conversely, assume that there exist sequences (g n ) n in L Q w (ν), and (h n ) n in L P ′ w −1 (µ) so that ∞ n=1 g n L Q w (ν) h n L P ′ w −1 (µ) < ∞, and for all f ∈ L P w (µ) g n (x)h n (y) f (y)dµ(y) , a.e x.

By Lemma 4.1 (d) we have
To prove that T f = n f, h n g n in L Q w (ν) we let s n := n j=1 f, h j g j , then (s n ) n is a sequence in L Q w (ν) and Moreover, γ is well defined and γ ∈ L Q w (ν) since it is the increasing limit of the sequence (γ n ) n = ( f L P w (µ) n j=1 h j L P ′ w −1 (µ) |g j (x)|) n of L Q w (ν) functions and By the Levi monotone convergence theorem we see that γ ∈ L Q w (ν). Finally, applying the Lebesgue dominated convergence theorem we deduce that s n → T f in L Q w (ν). Before establishing a characterisation of r-nuclear operators for more general measures and weights we first generalise Lemma 4.1. The following definition will be useful. Λ i . For a measure µ, a weight w on Ω and a multiindex P we will say that the triple (µ, w, P ) is σ-finite if there exists a family of disjoint boxes Ω k such that µ(Ω k ) < ∞, ∞ k=1 Ω k = Ω and w P (Ω k ) = 1 Ω k L P w (µ) < ∞.
Remark 4.5. We observe that for the case of a single measure space (l = 1), a triple (µ, w, p) is σ-finite if and only if wµ is σ-finite. If in addition we restrict to consider weights such that 0 < w(x) < ∞ then triple (µ, w, p) is σ-finite if and only if the measure µ is σ-finite.
Proof. (a) Since the triples (µ, w, P ), (ν, w −1 , Q ′ ) are σ-finite, there exist two sequences (Ω k ) k and (Ξ j ) j of disjoint subsets of Ω and Ξ respectively such that k Ω k = Ω, j Ξ j = Ξ and for all j, k We now consider the measure spaces (Ω k , M k , µ k ) and (Ξ j , M ′ j , ν j ) that we obtain by restricting Ω to Ω k , and Ξ to Ξ j for every k, j, and restricting the functions g n to Ξ j , and h n to Ω k . Then, for all k, j By Lemma 4.1 (a) it follows that ∞ j=1 g j (x)h j (y) converges absolutely for a.e (x, y) ∈ Ξ j × Ω k . Hence We can now formulate a characterisation of r-nuclear operators on weighted mixednorm spaces and a trace formula.
Theorem 4.7. Let 0 < r ≤ 1. Let (Ω i , M i , µ i )(i = 1, . . . , l), (Ξ j , M ′ j , ν j )(j = 1, . . . , m) be measure spaces. Let 1 ≤ P, Q < ∞. Let w, w be weights on Ω, Ξ respectively such that the triples (µ, w, P ), (ν, w −1 , Q ′ ) are σ-finite . Then T is rnuclear operator from L P w (µ) into L Q w (ν) if and only if there exist a sequence (g n ) in L Q w (ν), and a sequence (h n ) in L P ′ w −1 (µ) such that Moreover, if w = w satisfies (2.1), µ = ν, P = Q and T is r-nuclear on L(L P w (µ)) with r ≤ 2 3 , then where λ j (j = 1, 2, . . . ) are the eigenvalues of T with multiplicities taken into account, and Proof. Again, for the proof of the characterisation it is enough to consider the case r = 1. But that characterisation now follows from the same lines of the proof of Theorem 4.3 by replacing the references to part (d) of Lemma 4.1 by the part (d) of Lemma 4.6. On the other hand, since w additionally satisfies (2.1) the metric approximation property holds. If T is r-nuclear with r ≤ 2 3 on L P w , the trace formula follows from the aforementioned Grothendieck's theorem in the introduction.

r-nuclearity on modulation spaces and the harmonic oscillator
In this section we describe the r-nuclearity and a trace formula in modulation spaces. We restrict our attention to modulation spaces but note that the same conclusions hold also in the Wiener amalgam spaces. Thus, as an immediate consequence of Theorem 4.7 and Corollary 3.1 we have: Corollary 5.1. Let 0 < r ≤ 1, 1 ≤ p, q < ∞ and w a submultiplicative polynomially moderate weight. An operator T ∈ L(M p,q w , M p,q w ) is r-nuclear if and only if its kernel k(x, y) can be written in the form Moreover, if T is r-nuclear with r ≤ 2 3 , then where λ j (j = 1, 2, . . . ) are the eigenvalues of T with multiplicities taken into account.
In principle, in Corollary 5.1, the order r ≤ 2 3 in the r-nuclearity is sharp for the validity of the trace formula (5.1) in the context of general Banach spaces (with approximation property). However, for the traces in, for example, the L p -spaces the trace formula (5.1) may hold for r-nuclear operators also with larger values of r. In fact, the condition r ≤ 2 3 may be relaxed to the condition r ≤ r 0 (p) with the index r 0 (p) ≥ 2 3 depending on p, see Reinov and Laif [RL13], as well as the authors' paper [DR14b]. The same property may be expected also for general operators in L P weighted mixed-norm spaces and consequently in modulation spaces.
However, for some special operators the trace formula (5.1) may be valid for even larger values of r, for example even for simply nuclear operators (i.e. for r = 1).
We shall now consider an application of such nuclearity concepts to the study of the harmonic oscillator A = −∆ + |x| 2 on R d . We will consider in particular the modulation space M p,q s corresponding to the weight w(x, ξ) = (1 + |ξ|) s . If Aφ j ≡ (−∆ + |x| 2 )φ j = λ j φ j , the eigenvalues λ j can be enumerated in the form λ = λ (k) = d i=1 (2k i + 1), k = (k 1 , . . . , k d ) ∈ N d , see e.g. [NR10, Theorem 2.2.3]. For the corresponding sequence of orthonormal eigenfunctions φ j in L 2 (R d ), we can write with convergence in L 2 (R d ): Hence, formally the kernel of A can be written as We note that this can be justified by taking negative powers of the harmonic oscillator (−∆ + |x| 2 ) −N for N > 0 large enough so that we start with the decomposition for the corresponding kernel in the form More generally for functions of the harmonic oscillator, defined by F (−∆ + |x| 2 )φ j = F (λ j )φ j , j = 1, 2, . . . , (5.2) we have: Theorem 5.2. Let 0 < r ≤ 1, s ∈ R and 1 ≤ p, q < ∞. The operator F (−∆ + |x| 2 ) is r-nuclear on M p,q s (R d ) provided that with the absolutely convergent series.
Proof. The first part follows from Corollary 5.1. Moreover, while formula (5.4) can be expected from the general Grothendieck's theory (at least for r ≤ 2 3 ), in this case it follows in an elementary way due to the smoothness property of Hermite functions. Indeed, from (5.2), the integral kernel of F (−∆ + |x| 2 ) is given by k(x, y) = ∞ j=1 F (λ j )φ j (x)φ j (y).
(5.5) At the same time, we know that functions φ j are all smooth and fast decaying, implying in particular that φ j ∈ M p,q s (actually, this also follows from the assumption (5.3) holds with r = 1). Consequently, by (1.5), we obtain in view of the equality φ j , φ j M p,q s ,M p ′ ,q ′ −s = (φ j , φ j ) L 2 = 1.

The series in (5.4) converges absolutely in view of
is finite by the assumption. This completes the proof.