1. Introduction

1.1. Outline of the paper

Pseudo-multipliers and multipliers associated to the harmonic oscillator arise from the study of Hermite expansions for complex functions on R^{
n
} (see Thangavelu [^{23}], [^{24}], [^{25}], [^{26}] [^{27}], [^{28}], Epperson [^{11}] and Bagchi and Thangavelu [^{1}]). At the same time, we note that pseudo-multipliers are pseudo-differential operators on R^{
n
} in view of the quantization process developed by Ruzhansky and Tokmagambetov in [^{17}] and [^{18}] when the reference operator is the harmonic oscillator. In this note, we are interested in the membership of pseudo-multipliers associated to the harmonic oscillator (also called Hermite pseudo-multipliers) in the Schatten classes, *S _{r}
*(

*L*

^{2}) on

*L*

^{2}(R

^{ n }). With this paper we finish the classification of pseudo-multipliers in classes of

*r*-nuclear operators on

*L*-spaces (see Barraza and Cardona [

^{p}^{2}], [

^{3}]), which on

*L*

^{2}(R

^{ n }) coincide with the Schattenvon Neumann classes of order

*r*. Our main result is Theorem 1.1 where we establish some criteria in order that pseudo-multipliers belong to the classes

*S*(

_{r}*L*

^{2}), 0 <

*r*≤ 2. In order to present our main result we recall some notions. Let us consider the sequence of Hermite functions on R

^{ n },

where *x* = (*x*
_{1}, ・ ・ ・ , *x _{n}
*) ∈ R

^{ n },

*ν*= (

*ν*

_{1}, ・ ・ ・ ,

*ν*) ∈ N

_{n}^{ n }

_{0}, and

*Hν*(

_{j}*x*) denotes the Hermite polynomial of order

_{j}*ν*. It is well known that the Hermite functions provide a complete and orthonormal system in

_{j}*L*

^{2}(R

^{ n }). If we consider the operator

*L*= −Δ + |

*x*|

^{2}acting on the Schwartz space

*S*(R

^{ n }), where Δ is the standard Laplace operator on R

^{ n }, then we have the relation

*LΦ*=

_{ν}*λ*,

_{ν}Φ_{ν}*ν*∈ N

^{ n }

_{0}. The operator

*L*is symmetric and positive in

*L*

^{2}(R

^{ n }) and admits a self-adjoint extension

*H*whose domain is given by

So, for *f* ∈ Dom(*H*), we have

The operator *H* is precisely the quantum harmonic oscillator on R^{
n
} (see [^{15}]). The sequence {*f*ˆ(*Φ _{v}
*)} determines the Fourier-Hermite transform of

*f*, with corresponding inversion formula

On the other hand, pseudo-multipliers are defined by the quantization process that associates to a function *m* on R^{
n
} × N^{
n
}
_{0} a linear operator *T _{m}
* of the form

The function m on R^{
n
} × N*
^{n}
*

_{0}is called the symbol of the pseudo-multiplier

*T*. If in (5),

_{m}*m*(

*x, ν*) =

*m*(

*ν*) for all

*x*, the operator

*T*is called a multiplier. Multipliers and pseudomultipliers have been studied, for example, in the works [

_{m}^{1}], [

^{20}], [

^{21}], [

^{22}], [

^{23}], [

^{24}] (and references therein) principally by its mapping properties on

*L*spaces. In order that the operator

^{p}*T*:

_{m}*L*

^{2}(R

^{ n }) →

*L*

^{2}(R

^{ n }) belongs to the Schatten class

*S*(

_{r}*L*

^{2}), in this paper we provide some (sharp) conditions on the symbol

*m*.

1.2. Pseudo-multipliers in Schatten classes

By following *A*. Grothendieck [^{12}], we can recall that a linear operator *T*: *E* → *F* (*E* and *F* Banach spaces) is *r*-nuclear, if there exist sequences (*e′ _{n}
*)

_{ n∈N₀}in

*E′*(the dual space of

*E*) and (

*y*)

_{n}_{ n∈N₀}in F such that

The class of *r*−nuclear operators is usually endowed with the quasi-norm

In addition, when *E* = *F* is a Hilbert space and *r* = 1 (resp. *r* = 2), the definition above agrees with the concept of trace class operators (resp. Hilbert-Schmidt). For the case of Hilbert spaces *H*, the set of *r*-nuclear operators agrees with the Schatten-von Neumann class of order *r* (see Pietsch [^{13}], [^{14}]). We recall that a linear operator *T* on a Hilbert space *H* belong to the Schatten class of order *r*, *Sr*(*H*), if

where {*λ _{n}
*(

*T*)} denotes the sequence of singular values of

*T*, which are the eigenvalues of the operator √

*T*∗

*T*. It was proved in [

^{2}] that a multiplier

*T*, with symbol satisfying conditions of the form

_{m}

where {*I _{s}
*}

^{n}_{s}_{=0}is a suitable partition of N

^{ n }

_{0}, and

*α*

_{r}_{,p₁,p₂}(

*s, ν*) is a suitable kernel, can be extended to a

*r*-nuclear operator from

*L*

^{p}^{₁}(R

^{ n }) into

*L*

^{p}^{₂}(R

^{ n }). Although is easy to see that similar necessary conditions apply for pseudo-multipliers, the

*r*-nuclearity for these operators in

*L*-spaces was characterized in [

^{p}^{3}] by the following condition:

a pseudo-multiplier

*T*can be extended to a_{m}*r*-nuclear operator from*L*^{p}^{₁}into*L*^{p}^{₂}if and only if there exist functions*h*and_{k}*g*satisfying_{k}

If we consider *p*
_{1} = *p*
_{2} = 2, and a multiplier *T _{m}
*, the conditions above can be replaced by the following more simple one,

because the set of singular values of a multiplier *T _{m}
* consists of the elements in the sequence . The condition (10) characterizes the membership of pseudomultipliers in Schatten classes in terms of the existence of certain measurable functions. However, in this paper we provide explicit conditions on

*m*in order to guarantee that

*T*∈

_{m}*S*(

_{r}*L*

^{2}), because explicit conditions allow us to known information about the distribution of the spectrum of these operators. Our main result is the following theorem.

**Theorem 1.1.**
*Let T _{m} be a pseudo-multiplier with symbol m defined on* R

^{ n }× N

^{ n }

_{0}.

*Then we have:*

*T*_{m}is a Hilbert-Schmidt operator on L^{2}(R^{ n }),*i.e*.,*T*∈_{m}*S*^{2}(*L*^{2}),*if and only if*

*If T*∈_{m}is a positive operator, then T_{m}is trace class, i.e., T_{m}*S*_{1}(*L*^{2}),*if and only if*

In general, on a Banach space compact linear operators are bounded operators. Taking into account that Schatten-von Neumann classes on Hilbert spaces are families of compact operators, our main theorem gives conditions for the *L*
^{2}(R^{
n
})-continuity of pseudomultipliers. The problem of finding “satisfactory” conditions for the *L*
^{2}(R^{
n
})-boundedness of pseudo-multipliers remains open, and it was proposed by Bagchi and Thangavelu in [^{1}]; with our main result and the conditions proposed in Cardona and Barraza [^{3}], we solve partially such problem. However, Bagchi-Thangavelu’s problem will be “satisfactorily” solved in the work Cardona and Ruzhansky [^{4}].

1.3. Related works

Now, we include some references on the subject. Sufficient conditions for the *r*-nuclearity of spectral multipliers associated to the harmonic oscillator, but in modulation spaces and Wiener amalgam spaces, have been considered by J. Delgado, M. Ruzhansky and B. Wang in [^{8}], [^{9}]. The Properties of these multipliers in *L ^{p}
*-spaces have been investigated in the references S. Bagchi, S. Thangavelu [

^{1}], J. Epperson [

^{11}], K. Stempak and J.L. Torrea [

^{20}], [

^{21}], [

^{22}], S. Thangavelu [

^{23}], [

^{24}] and references therein. Hermite expansions for distributions can be found in B. Simon [

^{19}]. The r-nuclearity and Grothendieck-Lidskii formulae for multipliers and other types of integral operators can be found in [

^{7}], [

^{9}]. On Hilbert spaces the class of

*r*-nuclear operators agrees with the Schatten-von Neumann class

*S*(

_{r}*H*); in this context operators with integral kernel on Lebesgue spaces and, in particular, operators with kernel acting of a special way with anharmonic oscillators of the form

*E*= −Δ

_{a}_{ x }+ |

*x*|

^{ a },

*a*> 0, has been considered on Schatten classes on

*L*

^{2}(R

^{ n }) in J. Delgado and M. Ruzhansky [

^{10}]. A complete treatment for

*L*-boundedness and

^{p}*L*-compactness properties in terms of the Littlewood-Paley theory and the Hörmander condition will be considered in Cardona and Ruzhansky [

^{p}^{4}]. The proof of our results will be presented in the next section.

2. Pseudo-multipliers in Schatten-von Neumann classes

In this section we prove our main result for pseudo-multipliers *T _{m}
*. Our criteria will be formulated in terms of the symbols

*m*. First, let us observe that every pseudo-multiplier

*T*is an operator with kernel

_{m}*K*(

_{m}*x, y*). In fact, straightforward computation shows that

for every *f* ∈ *D*(R^{
n
}). We will use the following result (see J. Delgado [^{5}], [^{6}]).

**Theorem 2.1.**
*Let us consider* 1 ≤ *p*
_{1}, *p*
_{2} < ∞, 0 < *r* ≤ 1 *and let qi be such that*. *Let* (*X*
_{1}, *μ*
_{1}) *and* (*X*
_{2}, *μ*
_{2}) *be σ-finite measure spaces. An operator T*: *L ^{p}
*

^{₁}(

*X*

_{1},

*μ*

_{1}) →

*L*

^{p}^{₂}(

*X*

_{2},

*μ*

_{2})

*is r-nuclear if and only if there exist sequences*(

*g*)

_{n}

_{n}in L^{p}^{₂}(

*μ*

_{2}),

*and*(

*h*)

_{n}*in L*

^{p}^{₁}(

*μ*

_{1}),

*such that*

*for every f* ∈*L ^{p}
*

^{₁}(

*μ*

_{1}).

*In this case, if p*

_{1}=

*p*

_{2}(

*see Section 3 of*[

^{5}])

*the nuclear trace of T is given by*

Now, we prove our main theorem.

*Proof of Theorem 1.1*. Let us consider a pseudo-multiplier *T _{m}
*. By definition, T

_{m}is a Hilbert-Schmidt operator if and only if there exists an orthonormal basis {

*e*}

_{ν}_{ ν }of

*L*

^{2}(R

^{ n }) such that

In particular, if we choose the system of Hermite functions {*Φ _{ν}
*}, which provides an orthonormal basis of

*L*

^{2}(R

^{ n }), from the relation

*T*(

_{m}*Φ*) =

_{ν}*m*(

*x, ν*)

*Φ*we conclude that

_{ν}*T*is of Hilbert-Schmidt type, if and only if

_{m}

So, we have proved the first statement. Now, if we assume that *T _{m}
* is positive, then

*T*is of class trace if and only if there exists an orthonormal basis {

_{m}*e*}

_{ν}_{ ν }of

*L*

^{2}(R

^{ n }) such that

As in the first assertion, if we choose the basis formed by the Hermite functions, *T _{m}
* is of class trace if and only if

which proves the second assertion. Now, we will verify that (14) implies that *T _{m}
* ∈

*S*(

_{r}*L*

^{2}) for 0 <

*r*≤ 1. For this, we will use Delgado’s Theorem (Theorem 2.1) to the representation (16) of

*K*,

_{m}

So, *T _{m}
* ∈

*S*(

_{r}*L*

^{2}) if

where we have used that the *L*
^{2}−norm of every Hermite function *Φ _{ν}
* is normalised. In order to finish the proof, we only need to prove that (15) assures that

*T*∈

_{m}*S*(

_{r}*L*

^{2}) for 1 <

*r*< 2. This can be proved by using the following multiplication property on Schatten classes:

So, we will factorize *T _{m}
* as

where *H* is the harmonic oscillator. Let us note that the symbol of *A* = *T _{m}H^{σ}
* is given by

*a*(

*x*,

*ν*) =

*m*(

*x*,

*ν*)(2|

*ν*| +

*n*)

^{ σ }. So, from the first assertion,

*A*∈

*S*

_{2}(

*L*

^{2}) if and only if

In order to prove that *T _{m}
* ∈

*S*(

_{r}*L*

^{2}), in view of the multiplication property

we only need to prove that *H*
^{−σ
} ∈ *S _{p}
*(

*L*

^{2}) with . The symbol of

*H*

^{−σ }is given by

*a′*(

*ν*) = (2|

*ν*| +

*n*)

^{−σ }. By using the hypothesis we have that

because . So, we finish the proof. ☑

2.1. Trace class pseudo-multipliers of the harmonic oscillator

In order to determinate a relation with the eigenvalues of *T _{m}
* we recall the following result (see [

^{16}]).

**Theorem 2.2.**
*Let T*: *L ^{p}
*(

*μ*) →

*L*(

^{p}*μ*)

*be a r-nuclear operator as in*(6).

*If*,

*then*,

*where λ _{n}
*(

*T*),

*n*∈ N,

*is the sequence of eigenvalues of T with multiplicities taken into account*.

As an immediate consequence of the preceding theorem (or the classical Grothendieck-Lidskii Theorem), if *T _{m}
*:

*L*

^{2}(R

^{ n }) →

*L*

^{2}(R

^{ n }) is trace class (1-nuclear) then,

where *λ _{n}
*(

*T*),

*n*∈ N, is the sequence of eigenvalues of

*T*with multiplicities taken into account.

_{m}