Integr. Equ. Oper. Theory 84 (2016), 463-485 DOI 10.1007/s00020-015-2273-y Published online December 30, 2015 © The Author(s) This article is published with open access at Springerlink.com 2015

Integral Equations and Operator Theory

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Born-Jordan Pseudodifferential Calculus, Bopp Operators and Deformation Quantization

Maurice A. de Gosson and Franz Luef

Abstract. There has recently been a resurgence of interest in BornJordan quantization, which historically preceded Weyl's prescription. Both mathematicians and physicists have found that this forgotten quantization scheme is actually not only of great mathematical interest, but also has unexpected application in operator theory, signal processing, and time-frequency analysis. In the present paper we discuss the applications to deformation quantization, which in its traditional form relies on Weyl quantization. Introducing the notion of "Bopp operator" which we have used in previous work, this allows us to obtain interesting new results in the spectral theory of deformation quantization.

Mathematics Subject Classification. Primary 47G30; Secondary 35Q40, 65P10, 35S05, 42B10.

Keywords. Moyal product, Born-Jordan operators, Bopp quantization, Pseudodifferential operators.

1. Introduction

Deformation quantization is a popular framework for quantum mechanics among mathematical physicists. It was suggested by Moyal [33] and Groe-newold [26], and put on a firm mathematical ground by Bayen et al. [1,2]; later Kontsevich [29,30] extended the theory to Poisson manifolds. Roughly speaking, the idea is to "deform" classical (Hamiltonian) mechanics into quantum mechanics using a parameter (Planck's constant); this is achieved using the notion of "star product" or "Moyal product" of two functions on R2". The star product is defined in physics by the suggestive formula

the exponential in the right hand side (the "Janus operator") is understood as a power series, the arrows indicating the direction in which the derivatives act.

l5> Birkhäuser

This formula was proposed for the first time by Groenewold in his seminal work [26] in 1946. A rigorous definition is the following: denoting by ^ the Weyl correspondence between operators and symbols, assume that a,b € S'(R2") and let A ^^ a and B ^^ b (A is sometimes called the "Weyl

transform" of a). If the product C = AB is defined and C <—i c then, by definition, c = a*% b.

We have shown in previous work [17,18] that we have

a*% b = Ab (2)

where A is a pseudodifferential operator acting on distributions defined on R2"; formally

A = a(x + 2ihdp,p- 2ihdx). (3)

We have called A the "Bopp pseudodifferential operator" with symbol a; it is the Weyl operator on T*R2" = R2" x R2" with symbol

a(z,C) = a (x - 2CP,P+ 1 Cx) (4)

where (Zx, Zp) are viewed as the dual variables of (x,p). This reformulation of the star product in terms of pseudodifferential operators is very fruitful; not only does it allow the study of the generalized eigenvalues and eigenfunctions of "stargenvalue" problems using standard pseudodifferential techniques, but it also leads to interesting regularity results in various functional spaces. The main observation, which leads to the theme of the present paper, is that the whole procedure heavily relies on the Weyl pseudodifferential calculus. From a physical point of view, this means that we are privileging Weyl quantization; technically this choice has many advantages because Weyl quantization is the simplest and most austere of all quantizations: using Schwartz's kernel theorem one shows that every continuous linear operator A: S(R") —> S'(R") can be viewed as a Weyl operator, and the Weyl correspondence is uniquely

characterized by the property of symplectic covariance: if A ^ a then

.S'A.S'-1 -J—^ a o Sfor every metaplectic operator S € Mp(n) with projection S € Sp(n) (the symplectic group). However, in real life things are not always that simple. Just a couple of years before Weyl [38] defined the eponymous correspondence, Born and Jordan [7], elaborating on Heisenberg's 1925 "matrix mechanics" [27], proposed a quantization procedure having a firm physical motivation (conservation of energy); their approach culminated one year later in their famous "drei Manner Arbeit" [8] with Heisenberg. There are many good reasons to believe that the Born and Jordan quantization scheme is the right one in physics (Kauffmann [28]); in addition, some very recent work of Boggiatto and his collaborators [3-5] shows that the Wigner formalism corresponding to Born-Jordan quantization is much more adequate in signal analysis than the traditional Weyl-Wigner approach. It allows to damp the appearance of unwanted "ghost" frequencies in spectrograms; numerical experiments confirm these theoretical facts.

In [16] the first of the authors has studied the properties of Born-Jordan pseudodifferential calculus; in the present paper we go one step further, and reformulate deformation quantization in terms of this calculus.

Notation. We will write z = (x,p) where x e R" and p £ (R")* = K". Operators S(R") —> S'(R") are usually denoted by A, B,... while operators S(R2") —> S'(R2") are denoted by A,B,... The lower-case Greek letters ... stand for functions (or distributions) defined on R" while their uppercase counterparts ... denote functions (or distributions) defined on R2". The distributional bracket on R" is denoted by {-, ■) and that on R2" by ((■, ■)). We denote by

a = dpi A dxi + ■■■ + dp" A dxn the standard symplectic form on T*R" = R" x R"; in coordinates: a(z, z') = Jz ■ z' where J = i j is the standard symplectic matrix.

2. Bopp Operators and Born-Jordan Quantization

2.1. Born-Jordan Versus Weyl

Let us quickly review the Born-Jordan and Weyl quantizations of monomials xmp£j. In what follows, the capital letters Xj and Pj denote operators acting on some space of functions or distributions on Rd, and satisfying Born's commutation relations

[Xj ,Pj]= Xj Pj - Pj Xj = ih. (5)

For instance, in traditional quantum mechanics d = n and Xj is the operator of multiplication by xj while Pj = -ihdx., but there is no compelling reason for limiting ourselves to these operators. Keeping this in mind, the Weyl quantization of monomials is given by the rule

x?pj ^ E QpTXTPk (6)

while Born-Jordan quantization is given by

x?p]J j—^ Pj-k XmPk. (7)

The Weyl and Born-Jordan correspondences agree for all monomials which are at most quadratic, as well as for monomials of the type pjxm or p?xj. They are however different as soon as we have l > 2 and m > 2 (Turunen [37]). It turns out that both rules can be obtained from the t-correspondence, defined by

xmpj EE (T) (1 - t)kt*-kPtkXmPk (8)

k=0 ^ '

where t is a real number. The case t = 2 yields the Weyl correspondence (6). Integrating the t-correspondence over the interval [0,1] and using the formula

/V - t)kT

k -kd = k\(i - k)! dT = (* + 1)!

we get the Born-Jordan correspondence (7). Historically, things evolved the other way round: in [7] Born and Jordan were led to the eponymous correspondence (7) by a strict analysis of Heisenberg's [27] ideas. In their subsequent publication [8] with Heisenberg they showed that their constructions extend mutatis mutandis to systems with an arbitrary number of degrees of freedom.

In the general case one proceeds as follows (de Gosson [16]): let t be a real parameter, and define the T-pseudodifferential operator AT = OpT(a) with symbol a G S'(R2") as being the operator S(R") —> S'(R") with distributional kernel

Kt(x,y) = i2-1[a(TX + (1 - T0](x - y) where F-1 is the inverse Fourier transform in the second set of variables. This defines the so-called Shubin T-correspondence [34] A ^^ a by; it is easy to check that one recovers the correspondence (8) for monomials. For a G S(R2") and ^ G S(R") the more suggestive formula

AT^(x) = if e%p(x-y)a(Tx + (1 - t)y,p)^(y)dpdy (9)

J JR2"

holds (Shubin [34], §23), which can also be extended to more general settings.

The choice t = 1 leads to the usual Weyl operators: A = OpBJ(a) with

A^(x)=ll e % p(x-y)a( 2 (x + y),p)^(y)dpdy. (10)

The Born-Jordan pseudodifferential operator A = OpBJ(a) is obtained by averaging the Shubin operators AT over t G [0,1]:

A = OpBj(a) = f At dT; (11)

it is thus the operator S (Rn) —> S'(Rn) with kernel

Kbj(x, y) = F2-1[aBj(x, y,p), -)](x - y) where the symbol aBJ is defined by

aBj(x, y,p) = I a(Tx +(1 - T)y,p)dT. (12)

(Heuristically, the Weyl operator (10) is obtained by approximating the integral in (12) using the midpoint rule). One verifies by a direct calculation, that the correspondence A a reduces to the Born-Jordan rules (7) for polynomials x^pp^.

As already mentioned, the Born-Jordan and Weyl correspondences agree for all quadratic polynomials in the variables xj,pj. More generally

(de Gosson [16]) both quantizations are also identical for symbols arising from physical Hamiltonians of the type

HW = Em(Pj - Aj(x))2 + ^(x) (13)

where Aj and V are real Cfunctions. 2.2. Harmonic Analysis of ABJ

It is usual to write Weyl operators A = OpW(a) in the form of operator valued integrals

A = ()" f aa(zo)T(zo)dzo (14)

where aa is the "twisted symbol" of A:

aa(z) = Faa(z) = (^)" f e-*a(z'z"> a(z')dz' (15)

(Fa is called the "symplectic Fourier transform") and T(z0) is the Heisenberg-Weyl operator defined, for z0 e R2" by

T(z0)^(x) = e*(p0X-1 P0X0V(x - x0) (16)

for a function (or distribution) ^ on R". Similarly, the Shubin operator AT = OpT (a) can be written (de Gosson [16])

At = (2^1)" aa (z0)TT (z0)dz0 (17)

where TT (z0) is the modified Heisenberg-Weyl operator defined by

TT (z0) = e 2* (2t-1)P0X0 T(z0). (18)

Proposition 1. (i) The Born-Jordan operator Abj = OpBj(a) is given by

Abj = (2ik)" f aa(z0)B(z0)T(z0)dz0 (19)

■J Rn

where © is defined by

= sin(poxo/2h). (20)

p0x0/2h

(ii) The twisted Weyl symbol aW of Abj is given by the explicit formula

aW (z0) = aa (z0 )©(z0). (21)

(iii) The operator Abj is hence a continuous operator S(R") —> S'(R") for every a £ S'(R2").

Remark 2. Notice that ©(z) = sinc(px/2h) where sinc(t) = (sint)/t is the cardinal sine function familiar from signal analysis.

Proof. The statement (ii) immediately follows from formula (19) taking the representation (14) of Weyl operators into account. The proof of formula (19) goes as follows (cf. [16], Proposition 11): integrating both sides of the equality (17) with respect to the parameter t G [0,1] one gets

Abj = (2^) I (zo) TT (zo)dT dzo.

■J rn \J0 J

Now, in view of definition (18), for p0x0 = 0

^ %(zo)dT =(^j1 e2%(2t-1)p0X0d^j T(zo) sin(poxo/2^),

poxo/2h

-T(zo)

hence formula (19) (the formula holds by continuity for poxo = 0). (iii) Formula (21) implies that aW G S'(Rn) if aa G S'(Rn) because © G LTO(M2") n C™ (R2n). □

It immediately follows from formula (20) that since ©(zo) = 0 for all zo = (xo,po) such that poxo = 2Nnh for some integer N G Z we see that an arbitrary continuous operator A: S(Rn) —> S'(Rn) is not in general a BornJordan operator: every such operator A has indeed a twisted Weyl symbol aW in view of Schwartz's kernel theorem, but because of zeroes of we cannot in general expect the Eq. (21) to be solved for aa. This property of Born-Jordan operators really distinguishes them among all traditional pseudodifferential operators: the Born-Jordan "correspondence" is neither surjective, nor injec-tive. Keeping this caveat in mind, we will still write symbolically a A or A = OpBJ(a).

2.2.1. Composition and Adjoints of Born-Jordan Operators. Let A: S(Rn)

—> S'(Rn) and B: S(Rn) —> S(Rn) be two continuous operators; their product AB is well-defined, and its Weyl symbol can be explicitly determined in terms of those of A and B. In fact if A = OpW(a) and B = OpW(b) then AB = OpW(a*n b) where akn b is the Moyal product:

(a -- b)(z) = (^)2" ff e2%*(u'v)a(z + 1 u)b(z - 2v)dudv. (22)

There are several ways to rewrite this formula; performing elementary changes of variables we have

(akn b)(z) = (n-)2n ff e-$°(z-z'a(z')b(z")dz'dz" (23)

j jr4n

which is well-known in the literature. For our purposes, it will be more tractable to use the following formula, which gives the twisted symbol of the compose in terms of the twisted symbols of the factors:

(akn b)a(z)= f e2%<z'z,)a,a(z - z')ba(z')dz'. (24)

Proposition 3. Let A = OpBJ(a) and B = OpBJ(5) be two Born-Jordan pseudodifferential operators; we suppose that C = AB is defined as an operator l) —> S '(M™ ). (i) The Weyl symbol of C = AB is given by the formula

cW(z)= e2k°(z'z')aa(z - z')ba(z')©(z - z')©(z')dz' (25)

where © is defined by (20). (ii) If we can factorize cW as cW(z) = x(z)©(z) where x & S'(R") then x = ca with C = OpBj(c). (iii) The adjoint of A = OpBj(a) is A* = OpBj(a). In particular, the Born-Jordan operator A is formally self-adjoint if and only its symbol is real.

Proof. (i) Formula (25) is an immediate consequence of formulas (24) and of (21) since ca = a*% b. The statement (ii) follows, using again (21). (iii) The adjoint of the r-pseudodifferential operator AT = OpT(a) is A* = Op^T(a) (see [15,34]); it follows that

A* = f Op1_T (a)dr = f OpT (a)dr = OpBJ(a). □

Remark 4. Note that x, and hence c, are not uniquely defined by the relation cW(z) = x(z)©(z) since ©(z) = 0 for infinitely many values of z. On the other hand, it is not obvious that an arbitrary Weyl operator can be written as a Born-Jordan operator. That this is however the case has been proven recently in Cordero et al. [11] using techniques from distribution theory (the Paley-Wiener theorem).

3. Bopp Quantization of Born-Jordan Operators 3.1. Bopp Calculus

Setting v = zo, z + 2u = z' in the formula (23) and introducing the notation

T(z0)b(z) = e-*a(z'za)b(z - 1 zo) (26)

we can rewrite formula (22) as

a*n b(z) = (2-^) I aa(zo)T(zo)b(z)dzo. (27)

The restrictions of the operators T(z0): S'(M2n) —» S'(M2n) to L2(R2n) are unitary, and satisfy the same commutation relations

T(zo)T(zi) = e ^-(z°'zi)^(zl )T(zo) (28)

as the Heisenberg-Weyl operators. In [17] we have proven the following result:

Proposition 5. The Weyl symbol of the operator At : b 1—> a b is the distribution T G S'(M" x Rn) defined by

T(z, Z) = ^z - IJÇ) = a(x - IZp,P + iCx) (29)

where (z,C) G T*R2n, C =(Cx,Cp).

Xj = xj + 17dPj , Pj = pj - ~dXj . (30)

We now introduce the following elementary operators (called "Bopp shifts" following Bopp [6]; also see Kubo [31]) acting on phase space functions and distributions:

—dPj, pj = pj - —.

These operators satisfy Born's commutation relations (5), and we can thus define the extended quantization rule

x?pj - £ (i)pi-k Xmpk (31)

k=o ^ '

corresponding to (6)-(7), respectively. The Weyl and Born-Jordan symbols of

xj — 1ZP'j and pj + 2

Xj and Pj being, respectively, xj — 1 Zpj and pj + 2Zx,j formula (29) suggests

the notation

A = ^x + 1 ihOp,p — 2 ihdx)

used in the Sect. 1.

3.2. The Born-Jordan Starproduct

In the Born-Jordan case we would like to define Bopp quantization using a procedure extending the natural correspondence

xfp'i yp—x-mPk.

J 1 J ^ +1 z_/ j J J

induced by the monomial rule (7). We will proceed as follows: returning to formula (17) we define the phase-space t-operator by

A = (onft) I aa (zo)tt (zo )dzo (32)

where TT (zo) is defined in terms of the operator (26) by

TT (zo) = e 2% (2t-1)p0X0 T(zo). (33)

In analogy with formula (2) we now define the "Born-Jordan starproduct"

-r'bj:

Definition 6. Let a G S'(R2n). The Bopp-Born-Jordan (BBJ) operator with symbol a is the operator

Abj = OPbj(a): S(R2n) S'(R2n)

defined by the integral

Abj = I AtdT (34)

where AT is the pseudodifferential operator (32). Let b G S(R2n). We set

a k-'BJ b = ABjb. (35)

In view of formula (19) the BBJ operator has the explicit expression

Abj = (2nh)n i (z)©(z)T(z)dz (36)

where © e L™(R2n) n C™ (R2n) is given by (20). 3.3. The Functions AmbBJ and WigBJ

In what follows ((■, ■)) denotes the distributional bracket on R2n.

The Weyl correspondence between symbols and operators can be defined using the Wigner formalism. In fact, given a symbol a e S(R2n) one can show

(see e.g. [15], §10.1) that the operator A ^ a is the only operator such that (AV#)L2 = ((a, Wig(^))) (37)

where Wig(^, () is the cross-Wigner distribution (or function) of e S (Rn):

Wig(V>, ()(z) = (^)n f e-*^(x + 2y)f(x - 1 y)dy. (38)

Noting that (A^|()L2 = (A^,f) and that Wig(^,() e S(R2n) formula (37) allows to extend the definition of the operator A to the case where a e S'(R2n). In view of Plancherel's theorem we can rewrite (37) as

(AV#)L2 = ((aa, Fa Wig(^, f)v)) (39)

where f v(z) = f (—z). Since the symplectic Fourier transform of the cross-Wigner transform is the cross-ambiguity function [14,15,23]

Amb(^, ()(z) = ()n i e-*pyV(y + 1 x)f(y - 1 x)dy

we have

(A^|^)L2 = ((aa, Amb(^,f)v)). (40)

It turns out that we have similar formulas for Born-Jordan operators. We first recall [15, §8.3.1] that the symplectic Fourier transform defined by (15) is involutive: F2 = Id and satisfies the following variant of the Plancherel identity where ((., .)) denotes the scalar product on R2n:

((a,Fa b)) = (((Fa a)v,b)) = ((Fa a,bv)). (41)

Proposition 7. Let a e S'(R2n) and e S(Rn); we have

(ABJ^)l2 = ((aa, AmbBj(^, f)v)) (42)

(ABj^)L2 = ((a, WigBj(^,^))) (43)

where AmbBj(^, () and WigBj (^,f) are defined by

AmbBJ f) = Amb(^, ()© (44)

WigBj (V>, f) = Fa AmbBj ($, f) (45)

where ©a = Fa© is the symplectic Fourier transform of ©.

Proof. In view of formula (19) we have

(Abj= ()" f aa(z)e(z)(T(z№\4>)L*dz.

Now, a straightforward calculation [15, §9.1.1] using the explicit expression for the Heisenberg-Weyl operator T(z0) shows that

(T(z)V#)L2 = (2n*)n Amb(V>,4)(-z)

so that

(abj^\^)l2 = aa (z)©(z)Amb(^,4)(-z)dz

hence formula (42) since ©(—z) = ©(z). By the second equality in Plancherel's formula (41), we have, since Fa is involutive,

((aa, AmbBj(^)v)) = ((a,Fa(AmbBj(^, 4)v))v)

= ((a,Fa (AmbBj(V^))»

= ((a, WigBj(^,^)>)

which is formula (43). □

The symplectic Fourier transform satisfying the convolution formula

Fau * Fav = (2n*)n Fa (uv) we have

WigBj(^, 4) = (^)n Fa Amb(^, 4) * Fa©

hence the explicit formula

WigBj (V>,4) = ()n Wig(^,^) * ©ff. (46)

Remark 8. Due to formula (46) the modified Wigner function WigBJ ^ is an element of the Cohen class (Cohen [12,13], Grochenig [24]); as such it can be viewed as a probability quasi-distribution having a similar status as that of the usual Wigner function (it has, for instance the "right" marginal properties): for ^ G L1(Rn) n L2(Rn) we have

f WigBJ = Wx)|2

[ WigBJ = \ftp) |2.

The symbol of Born-Jordan operators are obtained by averaging the r-symbol over [0,1] (formula (12)). A similar procedure holds for WigBJ: for t G R and G 5(Rn) define the T-cross-Wigner transform

WigT(^,4)(z) = (^)n f e-*^(x + Ty)^(x - (1 - t)y)dy. (47)

One proves (Boggiatto et al. [3]) that WigT belongs to the Cohen class; in fact:

WigT4) = Wig(^, 4) * Fa©(T) (48)

where ©(T) is the function

~ , x 2n (i 2px \

©(T)(z) = ^T-TFexK*2T-rJ. (49)

We have [5,16]

WigBJ(^)= / WigT (^ftdr. (50)

As the usual cross-Wigner transform, WigT satisfies a Moyal identity (or "orthogonality relation" as it is sometimes called): Boggiatto et al. [5] have shown that

((WigT(V>,()| WigT(V,f )))L2 = (2^)n W)L2 (f|f')L2 (51)

for every t e R and for all functions W,(,( in L2(Rn). However, the Moyal identity does not hold for WigBj. Here is why: let Q(V,f) = Wig(V,() * 0 (0 e S'(Rn)) be an element of the Cohen class. The Moyal identity is satisfied if and only if the Fourier transform 0 of the Cohen kernel 0 satisfies |0(z)| = (2nh)n (Cohen [12,13]). In the Born-Jordan case the Fourier transform of the Cohen kernel is the function ©(z) = sinc(px/2nk) which does not satisfy this condition.

4. Intertwiners

We are going to show that the usual Born-Jordan operator ABj = OpBj(a) and the corresponding BBJ operator ABj = OpBj(a) are intertwined by a family of linear mappings L2 (Rn) —> L2(R2n). This important result will allow us to study the regularity and spectral properties of the BBJ operators.

Definition 9. For f e S(Rn) with ||(||L2 = 1 we denote by U$ and ) the linear operators L2(Rn) —> L2 (R2n) defined, by

U^ = (2nh)n/2 WigBj (V, () (52)

UHT= (2nh)n/2 WigT(V,f).

We will call and U$ (T) the Born-Jordan and T-intertwiner, respectively The reason for this terminology will become clear in a moment.

4.1. The Intertwining Property

Recall that we defined (formula (26)) the unitary operator T(z0) : L2(R2n) —> L2 (R2n) by

T(zo)V(z)= e-*- 1 zo). We will need the following property of the cross-Wigner transform: Lemma 10. We have

Wig(T(zo)V, () = TT(zo) Wig(V, (). (53)

Proof. The cross-Wigner transform has the following well-known transla-tional property ([23], [15], §9.2.2): for all z0,z1 G R2n

Wig(T(zo)^, T(zi)4)(z) = e-*Wig(^, 4) (z - 1 (zo + zi)z)

where the phase y is given by

Y(z, zo, zi) = a(z, zo - zi) + 20"(zo, zi).

Taking z1 = 0 yields

Wig(TT(zo)^, 4)(z) = e-*Wig(^, 4)(z - izo)

which is (53). □

The interest of the definition of the mapping U^ comes from their intertwining properties:

Proposition 11. Let ABJ = OpBJ(a) and ABJ = OpBJ(a). The following intertwining properties

AbjU0 = u^abj, U;Abj = abju; (54)

hold for all 4 G S(Rn).

Proof. Let ^ G S(R2n). In view of formula (36) we have

Abj*(z) = (^)n i aa(zo)©(zo)T^(zo)^(z)dz jr2"

and hence, for ^ = U^.

AbjU^(z) = n aa (zo)©(zo)T(zo)(U^)(z)dz.

We have

T(zo)(U^^)(z) = (2nh)n/2T(zo)WigBj(^,4)(z)

= (2nh)n/2T(zo)(Wig(^, 4) * ©a)(z). In view of formula (53) we have

T(zo)(Wig(^,4) * ©ff)(z)= / T(zo)Wig(^,4)(z - u)©a(u)du

= f Wig(T(zo)^,4)(z - u)©a(u)du

= WigBj(T(zo)^,4)

and hence

a4bjU0 ^(z) = (2^)n aa (zo )©(zo)(U0T(zo)^)(z)dz

= U0Abj^(Z)

which proves the first formula (54). The second formula follows from the equalities

uiabj = (AbjU0)* = (U0Abj)* = AbjU* □

4.2. Properties of Intertwiners

We begin by considering the r-intertwiners.

Proposition 12. (i) The r-intertwiner U;(T) is a linear isometry of L2(Rn) on a closed subspace H;(T) of L2(R2n). (ii) The adjoint U; (T) is given by the formula

u; (T)t(y) = (-k)n/2 i e-^p(y-x)$(2x - y)(t * Fa©(T))(x,p)dpdx (55)

where ©(T) is defined by (49).

Proof. (i) Taking $ = with ||$|| = 1 in Moyal's formula (51) we have

((U;,(t )^iU;,(t )v))u* = (V#') (56)

hence U; (T) is an isometry. By definition of the adjoint we have

((U;(t )V#)l = (.m;,(T ^h*.

Set P;,(t) = U;,(T)U;(T); we have P*(t) = P;,(t) and P;P;(t) = U;,(t) U; (T) = P;,(T) because U; (T)U;,(T) is the identity on L2(Rn). It follows that P;,(T) is the orthogonal projection on H;,(T); since the range of a projection is closed, so is H;,(T). (ii) By definition of U; (T) we have

((U;,(T) V|*))l* = (^)n/2 ((Wig(^, * Fa©(T) |^))L*. (57)

(cf. formula (46)). Recalling the classical formula (f * g|h) = (f Igv * h) and noting that Fa) = Fa©(T), the formula above becomes

((U; ,(T )V|*))L* = ( ^ )n/2 ((Wig(^,$)|^ * Fa ©(T )))L* .

Taking the definition (38) of Wig(^, into account, we get

((U;,(t)№))l> = (27!f2 [ + 1 y)

x[ e-%py$(x - 1 y)(* * Fa©(T))(x,p)dpdx\ dy \jR*n J

that is, setting u = x + 1 y,

2 )n/2

((U;,(T)^|*))L* = (& )n/2 i №

xU e-% p(u-x)$(2x - u)(^ * Fa ©(T ))(x,p)dpdx \ du

\JR*n J

U; (t)*(u) = (n!)n/2 f e-%p(u-x)$(2x - u)(* * Fa©(T))(x,p)dpdx which is formula (55). □

We would now like to extend this result to the intertwiners U$. However, the proof of part (i) of Proposition 12 relies on the Moyal identity (51), since the latter allows to derive (56). However, as we have remarked above, the Moyal identity does not hold for the transform WigBJ(^,^). We must thus expect a somewhat weaker result. We will need the following lemma, which is a kind of interpolation result:

Lemma 13. Let t and t/ be two real numbers and two windows M and M. There exists a constant C> 0 such that

| ((U4,,{rI U< C$v' 11 $| | | | | | (58)

for all ($, e L2(R") x L2(Rn).

Proof. This amounts to establishing the existence of a constant C$, v > 0 such that

|((WigT WigT, ($',4')))l2 | < (2nK)-nC$, $ uu w ||.

Using Cauchy-Schwarz's inequality we have

|((WigT WigT' ($',4')))l2 | < || WigT ($,M)|| || WigT, (^,4')||.

Applying Moyal's identity to the terms in the right-hand side we have

|| WigT($,M)|| = (2kT |MMM n WigT,= (^)n un un

hence the inequality (58) with = ||M|| HM||. D

Let us now prove the analogue of Proposition 12 for the Born-Jordan intertwiners:

Proposition 14. (i) The Born-Jordan intertwiner U$ is a continuous linear mapping L2(Rn) —> L2(R2n). (ii) The adjoint U$ is given by the formula

U$*(y) = (¿J)n/2 f e-%p(y-x)M(2x - y)(* * FaG)(x,p)dpdx. (59)

(iii) Let (Mj)jeF be an orthonormal basis of L2(Rn) and set $jk = U$jMk. The system (&jk)(j,k)eFxF spans L2(R2n).

Proof. (i) We have

((U0$|U0j$/))L2 = i u$(t )$dT

u0,(t'jj

) ) L2

((u0,(t)^\u0,(t')^))l2 •

[0,1] x [0,1]

In view of formula (58) we thus have

|((u^|u0y))L2\<C^\\^\\ U'\\

where C0 = C0,0, which proves the continuity of (ii) The proof of formula (59) is similar to the proof of (55) in Proposition 12. (iii) We have to show that if (($jfc|^))L2 = 0 for all (j, k) G F x F then ^ = 0 almost everywhere.

We have ((^|$))L2 = (Uj)L2 hence ((^|$jfc))L2 = 0 is equivalent to (Uj.^I4>k)L2 = 0; since this equality holds for all k G F it follows that U j^ = 0 (for all j G F). We have

((*jfc|*))L2 = ( 2nh T" ((Wig(^k, hj ) * ©, |*))L2

= (2ns )n/2 ((Wig(^fc )|* * ©. ))L2.

The family of functions (Wig(^k,hj))(j,k)eFxF being an orthonormal basis of L2(R2n) (de Gosson and Luef [17], Lemma 3), it follows that ^ * ©a = 0, and hence

Fa(* * ©a) = (2nh)-n (Fa*)© = 0.

Since the set of zeroes of the function © is the union of the null sets {z: px = 2Nnh} (N G Z) we have Fa^ = 0 a.e. and hence ^ = 0 a.e., which was to be proven. □

5. Functional and Symbol Spaces 5.1. The Modulation Spaces Mq(R™)

The theory of modulation spaces goes back to Feichtinger [20,21]; for a detailed exposition see Grochenig [24]. The traditional definition of these functional spaces makes use of the short-time Fourier transform (or Gabor transform) familiar from time-frequency analysis; we will replace the latter by the cross-Wigner transform whose symplectic symmetries are more visible; that both definitions are equivalent was proven in de Gosson and Luef [18] and de Gosson [15].

We will use the notation (z)s = (1 + |z|2)s/2 for z G R2n; here s is any nonnegative real number. It follows from Peetre's inequality that the function z i—> (z)s is submultiplicative:

{z + z)s < 2s{z)s{z')s. (60)

Let q be a real number > 1, or to. We denote by Lqs (R2n) the space of all Lebesgue-measurable functions ^ on R2n such that {-)s^ G Lqs (R2n). When q < to the formula

ML = (! KzWzWdz) \J r2n /

defines a norm on Lqs (R2n); when q = to we set

||^||L~ = esssup|{z)s^(z)|.

z£r2n

Let now h be a fixed element of S(Rn), hereafter to be called a "window". For q < to the modulation space Mq (Rn) is the vector space consisting of all ^ G S'(Rn) such that Wig(^, h) G Lqs (R2n) where by (38) is the cross-Wigner transform [14,15]; equivalently

UnM = (/ |{z)s Wig^hKz^dz)/ < to. (61)

The space M°°(Rn) is similarly defined by

$ = ess sup |(z)s Wig(m,M)(z)| < x>. (62)

One shows that in both cases the definitions are independent of the choice of the window M, and that the || • HM^ (1 < q < to) form a family of equivalent

norms on Mq (Rn), which becomes a Banach space for the topology thus defined; in addition Mq(Rn) contains S(Rn) as dense subspace.

The class of modulation spaces Mq (Rn) contain as particular cases many of the classical function spaces. For instance

M2(Rn) = L2(R2n) n Hs (R2n)

which is the Sobolev-like space Qs(R2n) studied by Shubin [34], p. 45. We also have

S(Rn) = p| M2(Rn).

A particularly interesting example of modulation space is obtained by choosing q =1 and s = 0; the corresponding space M0i(Rn) is often denoted by S0(Rn), and is called the Feichtinger algebra [21] (it is an algebra both for pointwise product and for convolution). We have the inclusions

S(Rn) c S0(Rn) c C0(Rn) n L1 (Rn) n L2(Rn). (63)

5.2. Metaplectic and Heisenberg-Weyl Invariance Properties

Recall that the Wigner transform and the Heisenberg-Weyl operators satisfy

Wig(T(z0)m,T(z0)M)(z) = Wig(m, M)(z - Z0). (64)

for all e S/(Rn). Let (R2n,a) be the standard symplectic space. We denote by Sp(n) be the symplectic group of (R2n, a): we have S e Sp(n) if and only if S is a linear automorphism of R2n such that S * a = a. Equivalently,

ST JS = SJST = J where J = ^-Oj^ The symplectic group has a unique

(connected) covering group of order two; the latter has a true representation as a group Mp(n) of unitary operators on L2(Rn); this group is called the metaplectic group. The covering projection n: Mp(n) —> Sp(n) is uniquely defined up to inner automorphisms; we calibrate this projection so that we have n( J) = J where .J e Mp(n) is the modified Fourier transform defined

Jm(x) = ( )n/2 e-inn/4 i e-%m(x/)dx/.

(we refer to [14,15,23] for detailed studies of the metaplectic representation). The modulation spaces Mq(Rn) have remarkable invariance properties:

Proposition 15. (i) Each space Mq(Rn) is invariant under the action of the Heisenberg-Weyl operators T(z); in fact there exists a constant C > 0 such that

mzmlq < c(z)smMq. (65)

(ii) For 1 < q < to the space Mq (R") is invariant under the action of the metaplectic group Mp(re): if S G Mp(n) then S^ G Mq (R") if and only if ^ G Mq(R"). In particular Mq(R") is invariant under the Fourier transform.

(See de Gosson and Luef [18], Grochenig [24]).

A remarkable property of the Feichtinger algebra is that it is the smallest Banach space invariant under the action of the Heisenberg-Weyl operators (16) and of the metaplectic group.

5.3. The Sjostrand Symbol Classes

In [35,36] Sjostrand introduced a class M1(R2") of general pseudodifferential symbols; Grochenig [25] showed that this class is identical to the weighted modulation space M^'1(R2n) when s = 0.

5.4. Definition and Main Properties

Let us set, for s > 0,

«z,C»s = (1 + |z|2 + IC I2)s /2. (66)

By definition, MO' 1(R2") consists of all a G S'(R2") such that there exists a function $ G S(R2") for which

f sup |Wig(a, $)(n,C)|«n,C»s<< to (67)

JR2» z£i2"

where Wig(a, $) is the cross-Wigner transform on R2":

Wig(a, $)(z, C) = (^ f" [ e-*a(z + ± z')$(z - 1 z')dz'. (68)

When s = 0 one obtains the Sjostrand class: MTO'1(R2") = M0oo'1(R2"). It is easy to check that for every window $ G S(R2") the formula

IMI^,! =i sup [|Wig(a, $)(z,C)|((z,C))s ]dC< to (69)

s Ji2" z£i2"

defines a norm on Msoo'1(R2"). As for the modulation spaces Mq(R") condition (69) is independent of the choice of window $, and when $ runs through S(R2") the functions || • form a family of equivalent norms on

M°'1(R2"). It turns out that Msoo'1(R2") is a Banach space for the topology defined by any of these norms; moreover the Schwartz space S(R2") is dense in M°'1(R2").

The Sjostrand classes MO' 1(R2") contain many of the usual pseudodifferential symbol classes and we have the inclusion

C2k+1(R2") c M0O'1(R2") (70)

where Cb;k+1(R2") is the vector space of all functions which are differentiable up to order 2n +1 with bounded derivatives. In fact, for every window $ there exists a constant C$ > 0 such that

||a||^-^,i < C$||a||G2fc+i = C$ Udaa||. (71)

\a\<2k + 1

We first recall the following result, which says that these space are invariant under linear changes of variables:

Proposition 16. Let M be a real invertible 2n x 2n matrix. If a G M°°'1(M2n ) then a o M G M°°'1(M2n), and there exists a constant Cm > 0 such that for every window $ and every a G M°°'1(R2n) we have

Il a o M\\Mr,i < Cm\\a\\MrA (72)

where ^ = $ o M-1

For a proof of this result, see Proposition 7 in de Gosson and Luef [18]. We are next going to show that M°°' 1(R2n ) is invariant under the action of the metaplectic group Mp(2n). Denoting by S the generic element of Mp(2n) we have:

Proposition 17. Let S G Mp(2n) and a G S'(M2n). We have a G M°°'1(M2n) if and only if Sa G M°°'1(M2n) and we have

\ \ Sa \ \ MMr,i < Amax ! ! Sa \ \ Mr,i (73)

where Amax is the largest eigenvalue of STS G Sp(2n), S = n(S').

Proof. Let S G Sp(2n) be the projection of S. We have

\ \ Sa \ \ MU = f supO Wig(Sa,S$)(z,C) \ {{(z,C))Y K

s jr2" z£r2"

= i sup[ \ Wig(a, $)(S-1 (z,Z))\{{(z,Z)))s]dZ

jr2" z£r2"

= i supO Wig(a, $)(z,Z)) \ {{S (z,()))s]d(.

jr2" z£r2"

Now {{S(z,Z))) < Amax{{z,Ç)) hence

\ \ Sa\\M-,1 < Amax I' sup [ \ Wig(a, $)(z,Z))\ {{z,0)s]dC

s Jr2" z£r2"

which is the inequality (73). □

5.5. Regularity Properties

The following result is well-known (see e.g. Grochenig [25]); it shows that the Weyl correspondence a ^ A is a continuous mapping M°°' 1(R2n) —>

Proposition 18. Let a G Ms00'1(M2n). The Weyl operator A ^^ a is bounded on Mq(Rn) for every q G [1, œ], and there exists a constant C > 0 independent of q such that following uniform estimate holds

I I A11 B(Mq) < CI I a11 Mr-1

for all a G MO' 1(K2n) ( | | • | | Mq is the operator norm on the Banach space

The Sjostrand class MO' 1(R2") contains the Hormander symbol class S|°'0(R2") consisting of all a G CO (R2") such that for every pair of multi-indices a, ¡3 G N" there exists Cap > 0 such that dad^a(x,p)| < Cap. The result above implies as a particular case a Calderon and Vaillancourt [9] type

result: if a G S° 0(R2") then A S a is bounded on L2(R").

For our purposes the following property is very important:

Proposition 19. Let a,b G MSOO'1(R2"). Then a*h b G M°'1(R2"). In particular, for every window of the type $ = Wig p where p G S(R"), there exists a constant C$ > 0 such that

||a*r b||M-i < C$||a||Ms-i ^Mr-i ■

Since obviously a G M°'1(R2") if and only if a G M°'1(R2") the property above can be restated by saying that MO' 1(R2") is a Banach *-algebra with respect to the Moyal product if and only if C$ < 1 and the involution a i—> a.

6. Spectral Properties of the BBJ Operators

Recall that intertwining properties (54) hold more generally:

Abj U; = Abj , u; Abj = Abj u;

hold for all $ in Feichtinger's algebra S°(R"). Feichtinger has shown that a kernel theorem holds for S°(R"), see [21]. Suppose ABJ is a mapping from S°(R") to its dual space S° (R"). Then there exists a K G S°(R2") such that

Abj$(x) = J K(x,y)$(y)dy.

In this section we want to discuss generalized eigenvectors and generalized eigenvalues for Bopp Born Jordan operators that map S°(R") to S°(R") based on the Gelfand triple (S°(R"), L2(R"), S°(R") as advocated by Feichtinger with various collaborators in a series of papers [10,22] and provide an example of a Banach Gelfand triple. More concretely, we have that S°(R") is continuously and densely embedded into L2(R") and the Hilbert space L2(R") is w*-continuously and densely embedded into S°(R").

Note that the scalar product (., )L2 on L2(R") extends in a natural way to a duality between S°(R") and S° (R").

The usefulness of the Gelfand triple (S° (R"), L2(R"),S^ (R")) lies in the treatment of generalized eigenvectors for operators mapping Feichtinger's algebra into its dual space. As motivation for the notion of generalized eigenvectors we consider the translation operator Txf (t) = f (t — x) on S°(R"). Then the eigenvectors of Tx are given by the exponentials (t) = e2nlut for the eigenvalues e-2nlux for every w G R", but the eigenvectors are not in S°(R"). One way to cope with this problem, is to interpret the eigenvalue problem in a weak sense, see Maurin [32]. In our situation we have the following result, see [19]:

Lemma 20. Suppose A is a self-adjoint operator on So(M"). Then there exists a complete family of distributions (^a)aeA in SO(M") (the so-called generalized eigenvectors of A) such that

, A$) = Aa(^a, $) for each $ e So(M")

and that there exists a least one ^a such that $) = 0 for each $ e So(M"), and the generalized eigenvalues Aa of A.

Based on this useful fact we are going to treat spectral properties for Bopp Born Jordan operators. The Banach-Gelfand triple (S0(R"),L2(R"), SO (M")) provides a convenient setting for extending eigenvalue problems from the Hilbert space setting to a distributional framework.

Proposition 21. ¡Suppose Abj is an essentially self-adjoint operator from So(M") to SO(K"), i.e. the symbol of Abj is real-valued. (i) There exists a complete family of generalized eigenvectors }a£A and generalized eigenvectors {Aa}a£A for Abj with ^a in SO(K") for each a e A. (ii) Furthermore, Abj : So(R2") —> SO(K2") has a complete set {^}aeA of generalized eigenvectors with respect to generalized eigenvalues {Aa}a£A-

Proof. By assumption the operators ABJ has a complete family of eigenvectors by the preceding lemma if one considers the operator ABJ on S0(R") and one extends it to SO (K"). The correspondence between the eigenvectors of ABJ and ABJ follows from the intertwining relations (54) that extend naturally to this setting. If is a generalized eigenvector of ABJ, then ^a = is a generalized eigenvector of ABJ for the same eigenvalue. Suppose on the other hand, that ^a is a generalized eigenvector of ABJ, then U^ ^a is an eigenvector of ABJ corresponding to the same eigenvalue. □

Acknowledgments

Maurice de Gosson has been supported by a grant from the Austrian Research Fund FWF (Projektnummer P 27773-N25). Part of this work was done during a stay of MdG in the CIRM (Marseilles).

Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/ by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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Maurice A. de Gosson (b) NuHAG, Fakultät fUr Mathematik Universität Wien Oskar-Morgenstern-Platz 1 1090 Vienna Austria

e-mail: maurice.de.gosson@univie.ac.at

Franz Luef

Department of Mathematics

Norwegian University of Science and Technology

7041 Trondheim

Norway

e-mail: franz.luef@math.ntnu.no

Received: April 13, 2015. Revised: November 24, 2015.