Quantum white noise convolution operators with application to differential equations

Abstract

In this paper we introduce a quantum white noise
(QWN) convolution calculus over a nuclear algebra of
operators. We use this calculus to discuss new solutions of some
linear and non-linear differential equations.

Full text

QWN-Convolution operators with application to
differential equations
Abdessatar Barhoumi
Carthage University, Tunisia
Nabeul Preparatory Engineering Institute
Department of Mathematics
Campus Universitaire - Mrezgua - 8000 Nabeul
E-mail: abdessatar.barhoumi@ipein.rnu.tn
Alberto Lanconelli
Dipartimento di Matematica
Universita’ degli Studi di Bari
Via E. Orabona, 4
70125 Bari - Italia
E-Mail: lanconelli@dm.uniba.it
and
Hafedh Rguigui
Department of Mathematics
Faculty of Sciences of Tunis
University of Tunis El-Manar, 1060 Tunis, Tunisia
E-Mail: hafedh.rguigui@yahoo.fr
Abstract
In this paper we introduce a quantum white noise (QWN) convolution calculus
over a nuclear algebra of operators. We use this calculus to discuss new solutions
of some linear and non-linear differential equations.
Keywords: QWN-convolution operator, QWN-convolution product, QWN-derivatives, Wick
product.
Mathematics Subject Classification (2000): primary 60H40; secondary 46A32,
46F25, 46G20.
1 Introduction
In mathematics and in particular in functional analysis the convolution of two func-
tions is a new function that is typically viewed as a modified version of one of the original
functions. In infinite dimensional complex analysis [5], a convolution operator on the
test space Fθis a continuous linear operator from Fθinto itself which commutes with
all the translation operators, where the translation operator is defined by (see [3])
τ−xϕ(y) = ϕ(x+y), x, y ∈N0, ϕ ∈ Fθ.(1.1)
1

It is well known that τ−xis a continuous linear operator from Fθinto itself. The convo-
lution product of a distribution Φ ∈ F∗
θwith a test function ϕ∈ Fθis defined by (see
[3])
(Φ ∗ϕ)(x) = hhΦ, τ−xϕii, x ∈N0.(1.2)
It was proved in [6] that Cis a convolution operator on Fθif and only if there exists
Φ∈ F∗
θsuch that
C(ϕ) = Φ ∗ϕ, ∀ϕ∈ Fθ.(1.3)
The convolution operator Cis denoted by CΦ. It is noteworthy that the composition
CΦ1◦CΦ2is also a convolution operator so that there exists a unique element in F∗
θ
denoted by Φ1∗Φ2such that
CΦ1◦CΦ2=CΦ1∗Φ2,Φ1,Φ2∈ F∗
θ.(1.4)
The distribution Φ1∗Φ2defined in (1.4) is called the convolution product of Φ1and Φ2;
moreover
L(Φ1∗Φ2) = L(Φ1).L(Φ2),
where Lis the Laplace transform. Recently, in Ref. [16] it was shown that the notion of
convolution product of two white noise distributions (given via equality (1.4)) coincides
with their Wick product Φ1Φ2which is defined via the S-transform as
S(Φ1Φ2) = S(Φ1)S(Φ2).
To each Φ1we associate the Wick multiplication operator M
Φ1as
M
Φ1Φ2= Φ1Φ2,Φ2∈ F∗
θ.
Then we have that
(CΦ1)∗=M
Φ1,Φ1∈ F∗
θ.(1.5)
The Wick product of two white noise operators Ξ1, Ξ2∈ L(Fθ,F∗
θ), denoted by Ξ1Ξ2is
defined via the Wick symbol transform σto be the unique white noise operator satisfying
σ(Ξ1Ξ2) = σ(Ξ1)σ(Ξ2).
It is natural to ask what is the QWN analogue of the convolution of two white noise
operators. More precisely, what is the QWN analogue of (1.3) and (1.4)?
In this paper we introduce a new QWN-convolution calculus over a suitable nuclear
sub-algebra of L(Fθ,F∗
θ) using an adequate duality pairing hhh·,·iii and a new concept of
QWN-translation operator. Moreover, application to some differential equations is worked
out.
The paper is organized as follows. In Section 2, we summarize the common nota-
tions and concepts used throughout the paper. In Section 3, we give the analogue of
(1.1), we define the QWN-translation operator TQusing the Wick symbol map and we
give its integral representation with respect to QWN-coordinate systems {D±
s,(D±
t)∗}. In
Section 4, we define the QWN-convolution product an operator in L(F∗
θ,Fθ) and another
in L(Fθ,F∗
θ) as a QWN analogue of (1.2). In the same section an operator CQwhich
commutes with all QWN-translations operator TQis called QWN-convolution operator. We
characterize all operators CQand their integral representation is given. Finally, in Sec-
tion 5, we study a linear and non-linear differential equation associated to CQ.
2

2 Preliminaries
In this section we summarize the common notations and concepts used throughout
the paper.
2.1 Basic Gel’fand triples
Let Hbe the real Hilbert space of square integrable functions on Rwith norm |·|0
and E≡ S(R) be the Schwarz space consisting of rapidly decreasing C∞-functions.
Then, the Gel’fand triple
S(R)⊂L2(R, dx)⊂ S0(R) (2.1)
can be reconstructed in a standard way (see Ref. [14]) by the harmonic oscillator A=
1+t2−d2/dt2and H. The eigenvalues of Aare 2n+2, n = 0,1,2,· · · , the corresponding
eigenfunctions {en;n≥0}form an orthonormal basis for L2(R). In fact (en) are the
Hermite functions and therefore each enis an element of E. The space Eis a nuclear
space equipped with the Hilbertian norms
|ξ|p=|Apξ|0, ξ ∈E, p ∈R
and we have
E= projlim
p→∞ Ep, E0= indlim
p→∞ E−p,
where, for p≥0, Epis the completion of Ewith respect to the norm | · |pand E−pis
the topological dual space of Ep. We denote by N=E+iE and Np=Ep+iEp,p∈Z,
the complexifications of Eand Ep, respectively.
Throughout the paper, we fix a Young function θ, i.e. a continuous, convex and in-
creasing function defined on R+satisfying the two conditions: θ(0) = 0 and limx→∞ θ(x)/x =
+∞. The polar function θ∗of θ, defined by
θ∗(x) = sup
t≥0
(tx −θ(t)), x ≥0,
is also a Young function. For more details , see Refs. [7] and [15]. For a complex
Banach space (B, k·k), let H(B) denotes the space of all entire functions on B, i.e.
of all continuous C-valued functions on Bwhose restrictions to all affine lines of Bare
entire on C. For each m > 0 we denote by Exp(B, θ, m) the space of all entire functions
on Bwith θ−exponential growth of finite type m, i.e.
Exp(B, θ, m) = nf∈ H(B); kfkθ,m := sup
z∈B
|f(z)|e−θ(mkzk)<∞o.
The projective system {Exp(N−p, θ, m); p∈N, m > 0}and the inductive system
{Exp(Np, θ, m); p∈N, m > 0}give the two spaces
Fθ(N0) = projlim
p→∞;m↓0Exp(N−p, θ, m),Gθ(N) = indlim
p→∞;m↓0Exp(Np, θ, m).(2.2)
In the remainder of this paper we simply use Fθfor Fθ(N0). It is noteworthy that, for
each ξ∈N, the exponential function
eξ(z) := ehz,ξi, z ∈N0,
3

belongs to Fθand the set of such test functions spans a dense subspace of Fθ.
The space of linear continuous operators from Fθinto its topological dual space F∗
θ
is denoted by L(Fθ,F∗
θ) and assumed to carry the bounded convergence topology.
Let µbe the standard Gaussian measure on E0uniquely specified by its characteristic
function
e−1
2|ξ|2
0=ZE0
eihx,ξiµ(dx), ξ ∈E.
In all the remainder of this paper we assume that the Young function θsatisfies the
following condition
lim sup
x→∞
θ(x)
x2<+∞.(2.3)
It is shown in Ref. [7] that, under this condition, we have the nuclear Gel’fand triple
Fθ⊂L2(E0, µ)⊂ F∗
θ.(2.4)
Moreover, we observe that the spaces L(Fθ,Fθ), L(F∗
θ,Fθ) and L(L2(E0, µ), L2(E0, µ))
can be considered as subspaces of L(Fθ,F∗
θ). Furthermore, identified with its restriction
to Fθ, each operator Ξ in the nuclear algebra L(F∗
θ,F∗
θ) will be considered as an element
of L(Fθ,F∗
θ), so that we have the continuous inclusions
L(F∗
θ,F∗
θ)⊂ L(Fθ,F∗
θ),L(F∗
θ,Fθ)⊂ L(Fθ,F∗
θ).
2.2 Wick symbol and useful topological isomorphisms
The Wick symbol of Ξ ∈ L(Fθ,F∗
θ) is by definition ([14]) the C-valued function on
N×Nobtained as
σ(Ξ)(ξ, η) = hhΞeξ, eηiie−hξ,ηi, ξ, η ∈N, (2.5)
where hh·,·ii denotes the duality between the two spaces F∗
θand Fθ. By a density
argument, every operator in L(Fθ,F∗
θ) is uniquely determined by its Wick symbol. In
fact, if Gθ∗(N⊕N) denotes the nuclear space obtained as in (2.2) by replacing Npwith
Np⊕Np, we have the following characterization theorem for operator Wick symbols.
Theorem 2.1 ( See Ref. [12]) The Wick symbol map yields a topological isomorphism
between L(Fθ,F∗
θ)and Gθ∗(N⊕N).
Let Hθ(N⊕N) denote the restriction of the space Fθ(N0⊕N0) over N⊕N, i.e.
Hθ(N⊕N) = \
p≥0,γ1,γ2>0
Exp(Np⊕Np, θ, γ1, γ2),
where Exp(Np⊕Np, θ, γ1, γ2) denotes the space of all entire functions on Np×Npsuch
that
sup
(x1,x2)∈(Np×Np)
|g(x1, x2)|e−θ(γ1|x1|p)−θ(γ2|x2|p)<∞.
In other words, all holomorphic functions gin Hθ(N⊕N) admit a Taylor expansion
g(x1, x2) = Pl,mhgl,m , x⊗l
2⊗x⊗m
1ifor x1, x2∈N, where gl,m ∈(N⊗(l+m))sym(l,m)is such
that for all p∈Nand γ1,γ2>0
k−−→
σ(Ξ)k2
θ,p,(γ1,γ2):=
∞
X
l,m=0
(θlθm)−2γ−l
1γ−m
2|gl,m|2
p<∞.
4

Here θn= infr>0eθ(r)
rn, for n∈N. Then, using the kernel theorem and the reflexivity of
the space Fθ, we obtain an analogue of Theorem 2.1.
Proposition 2.2 ([2]) The Wick symbol map realizes a topological isomorphism between
the space L(F∗
θ,Fθ)and the space Hθ(N⊕N).
2.3 QWN-Derivatives
A typical element of the nuclear algebra L(Fθ,F∗
θ), that will play a key role in
our development, is the Hida’s white noise operator at. For z∈N0and ϕ(x) with
Taylor expansion P∞
n=0hx⊗n, fniin Fθ, the holomorphic derivative of ϕat x∈N0in the
direction zis defined by
(a(z)ϕ)(x) := lim
λ→0
ϕ(x+λz)−ϕ(x)
λ.(2.6)
We can check that the limit always exists, a(z)∈ L(Fθ,Fθ) and a∗(z)∈ L(F∗
θ,F∗
θ),
where a∗(z) is the adjoint of a(z), i.e. for Φ ∈ F∗
θand φ∈ Fθ,hha∗(z)Φ, φii =hhΦ, a(z)φii.
If z=δt∈E0we simply write atinstead of a(δt) and the pair {at, a∗
t}will be referred
to as the quantum white noise process. In QWN-field theory atand a∗
tare called the
annihilation and creation operators at the point t∈R.
It is a fundamental fact in quantum white noise theory [14] (see also Ref. [12]) that
every white noise operator Ξ ∈ L(Fθ,F∗
θ) admits a unique Fock expansion
Ξ =
∞
X
l,m=0
Ξl,m(κl,m ),(2.7)
where, for each pairing l, m ≥0, κl,m ∈(N⊗(l+m))0
sym(l,m)and Ξl,m (κl,m) is the integral
kernel operator characterized via the Wick symbol transform by
σ(Ξl,m(κl,m ))(ξ, η) = hκl,m, η⊗l⊗ξ⊗mi, ξ, η ∈N. (2.8)
This can be formally rewritten as
Ξl,m(κl,m ) = RRl+mκl,m(s1,· · · , sl, t1,· · · , tm)
a∗
s1· · · a∗
slat1· · · atmds1· · · dsldt1· · · dtm.
In this way Ξl,m(κl,m) can be considered as the operator polynomials of degree l+m
associated to the distribution κl,m ∈(N⊗(l+m))0
sym(l,m)as coefficient; and therefore every
white noise operator is a “function” of the quantum white noise. This gives a natural idea
for defining the derivatives of an operator Ξ ∈ L(Fθ,F∗
θ) with respect to the quantum
white noise coordinate system {at, a∗
t;t∈R}.
From Refs. [9], [10] and [11], we summarize the novel formalism of quantum white
noise derivatives. For ζ∈N,a(ζ) extends to a continuous linear operator from F∗
θinto
itself (denoted by the same symbol) and a∗(ζ) (restricted to Fθ) is a continuous linear
operator from Fθinto itself. Thus, for any white noise operator Ξ ∈ L(Fθ,F∗
θ), the
commutators
[a(ζ),Ξ] = a(ζ)Ξ −Ξa(ζ),[a∗(ζ),Ξ] = a∗(ζ)Ξ −Ξa∗(ζ),
5

are well defined white noise operators in L(Fθ,F∗
θ). The quantum white noise derivatives
are defined by
D+
ζΞ = [a(ζ),Ξ] , D−
ζΞ = −[a∗(ζ),Ξ].(2.9)
These are called the creation derivative and annihilation derivative of Ξ, respectively.
It is apparent that it is not straightforward to define D±
zfor each z∈N0because
the compositions a(z) Ξ and Ξ a(z) are not well defined in general and (2.9) makes no
sense if a(ζ), ζ ∈N, is replaced by a(z), z ∈N0. In Ref. [1], for z∈N0, it is shown that
D+
zis a continuous operator from L(Fθ,Fθ) into itself and D−
zis a continuous operator
from L(F∗
θ,F∗
θ) into itself. The pointwisely defined quantum white noise derivatives
D±
t≡D±
δtare discussed in Refs. [9], [10] and [11].
For any x, y ∈N0, we define the operator exponential Ξx,y by
Ξx,y ≡
∞
X
l,m=0
Ξl,m(κl,m (x, y)) ∈ L(Fθ(N0),F∗
θ(N0)),
where κl,m(x, y) = 1
l!m!x⊗l⊗y⊗m. It is noteworthy that {Ξx,y;x, y ∈N0}spans a
dense subspace of L(Fθ(N0),F∗
θ(N0)) and {Ξx,y;x, y ∈N}spans a dense subspace of
L(F∗
θ(N0),Fθ(N0)).
3QWN-translation operators
We start by clarifying the topology of the nuclear algebras L(Fθ,F∗
θ) and L(F∗
θ,Fθ).
From Theorem 2.1 and Proposition 2.2, we have the topological isomorphism:
L(Fθ,F∗
θ)' Gθ∗(N⊕N) = [
p≥0,γ1,γ2>0
Exp(Np⊕Np, θ∗,(γ1, γ2))
L(F∗
θ,Fθ)' Hθ(N⊕N) = \
p≥0,γ1,γ2>0
Exp(Np⊕Np, θ, (γ1, γ2)).
For p≥0 and γ1, γ2>0, let Lθ,−p,(γ1,γ2)(Fθ,F∗
θ) denotes the subspace of all Ξ ∈
L(Fθ,F∗
θ) which correspond to elements in Exp(Np⊕Np, θ∗,(γ1, γ2)). Similarly, let
Lθ,p,(γ1,γ2)(Fθ,F∗
θ) denotes the subspace of all Ξ ∈ L(F∗
θ,Fθ) which correspond to ele-
ments in Exp(Np⊕Np, θ, (γ1, γ2)).
The topology of Lθ,−p,(γ1,γ2)(Fθ,F∗
θ) is naturally induced from the norm of the Banach
space Exp(Np⊕Np, θ∗,(γ1, γ2)) which will be denoted by ||| · |||θ,−p,(γ1,γ2), i.e., for Ξ ∈
Lθ,−p,(γ1,γ2)(Fθ,F∗
θ),
|||Ξ|||θ,−p,(γ1,γ2)=kσ(Ξ)kθ∗,−p,(γ1,γ2)= sup
ξ,η∈Np
|σ(Ξ)(ξ, η)|e−θ∗(γ1|ξ|p)−θ∗(γ2|η|p).
Similarly, the topology of Lθ,p,(γ1,γ2)(F∗
θ,Fθ) is naturally induced from the norm of the
Banach space Exp(Np⊕Np, θ, (γ1, γ2)) which will be denoted by ||| · |||θ,p,(γ1,γ2), i.e., for
Ξ∈ Lθ,p,(γ1,γ2)(F∗
θ,Fθ),
|||Ξ|||θ,p,(γ1,γ2)=kσ(Ξ)kθ,p,(γ1,γ2)= sup
ξ,η∈Np
|σ(Ξ)(ξ, η)|e−θ(γ1|ξ|p)−θ(γ2|η|p).
For Ξ ∈ L(F∗
θ,Fθ), we define the operator T−c,−dby
T−c,−dσ(Ξ)(ξ, η) = σ(Ξ)(ξ+c, η +d), c, d, ξ, η ∈N. (3.1)
6

Proposition 3.1 Let Ξ∈ L(F∗
θ,Fθ). Then T−c,−d(σ(Ξ)) ∈ Hθ(N⊕N). Moreover, for
all α≥0and γ1, γ2>0, there exist α0≥αand γ0
1,γ0
2∈]0,1[ such that
kT−c,−dσ(Ξ)kθ,α,(γ1,γ2)≤eθ(M1|c|α0)eθ(M2|d|α0)|||Ξ|||θ,α0,(γ0
1,γ0
2)
with M1=γ0
1
1−γ0
1and M2=γ0
2
1−γ0
2.
Proof. By definition for any α0≥0, γ0
1, γ0
2>0, we have
|T−c,−dσ(Ξ)(ξ, η)| ≤ |||Ξ|||θ,α0,(γ0
1,γ0
2)eθ(γ0
1|ξ+c|α0)+θ(γ0
2|η+d|α0).
Then, for 0 < γ0
1<1, 0 < γ0
2<1, α≥0, γ1>0 and γ2>0, the convexity of θyields
|T−c,−dσ(Ξ)(ξ, η)|e−θ(γ1|ξ|α)e−θ(γ2|η|α)
≤ |||Ξ|||θ,α0,(γ0
1,γ0
2)eγ0
1θ(|ξ|α0)e(1−γ0
1)θ(γ0
1
1−γ0
1
|c|α0)e−θ(γ1|ξ|α)
×eγ0
2θ(γ0
2|η|α0)e(1−γ0
2)θ(γ0
2
1−γ0
2
|d|α0)e−θ(γ2|η|α).
We choose α0> α,γ1≥ kiα0,αkH S and γ2≥ kiα0,αkH S , to get
eθ(|ξ|α0)e−θ(γ1|ξ|α)≤1, eθ(|η|α0)e−θ(γ2|η|α)≤1.
This gives
kT−c,−dσ(Ξ)kθ,α,(γ1,γ2)≤eθ(M1|c|α0)eθ(M2|d|α0)|||Ξ|||θ,α0,(γ0
1,γ0
2)
with M1=γ0
1
1−γ0
1,M2=γ0
2
1−γ0
2, 0 < γ0
1<1, 0 < γ0
2<1 and α0> α such that kiα0,αkH S ≤
min(γ1, γ2). The holomorphy of σ(Ξ) completes the proof.
From Proposition 3.1 and Proposition 2.2 we introduce the QWN-translation operator
as follows.
Definition 3.2 The QWN-translation operator is defined by
TQ
−c,−dΞ = σ−1(T−c,−d(σ(Ξ))),Ξ∈ L(F∗
θ,Fθ).(3.2)
From (3.1) and (3.2) it is clear that TQ
−c,−dis a continuous linear operator from
L(F∗
θ,Fθ) into itself.
Theorem 3.3 The QWN-translation operator admits the following integral representation
on L(F∗
θ,Fθ):
TQ
−c,−d=
∞
X
l,m=0
1
l!m!ZRl+m
d(s1)· · · d(sl)c(t1)· · · c(tm)D+
s1· · · D+
slD−
t1· · · D−
tmds1· · · dsldt1· · · dtm·
(3.3)
7

Proof. Let Ξ = P∞
l,m=0 Ξl,m(κl,m)∈ L(F∗
θ,Fθ). Then
T−c,−dσ(Ξ)(ξ, η)
=σ(Ξ)(ξ+c, η +d) (3.4)
=
∞
X
l,m=0
hκl,m,(η+d)⊗l⊗(ξ+c)⊗mi
=
∞
X
i,j,l,m=0
(l+i)!(m+j)!
i!j!l!m!Dd⊗lb
⊗lκl+i,m+jb
⊗mc⊗m, η⊗i⊗ξ⊗jE.(3.5)
On the other hand, it is a fact that (see [9] and [2])
D−
zΞl,m(κl,m ) = mΞl,m−1(κl,m b
⊗1z) (3.6)
and
D+
zΞl,m(κl,m ) = lΞl−1,m(zb
⊗1κl,m),(3.7)
where, for zp∈(N⊗p)0, and ξl+m−p∈N⊗(l+m−p),p≤l+m, the contractions zpb
⊗pκl,m
and κl,m b
⊗pzpare defined by
hzpb
⊗pκl,m, ξl−p+mi=hκl,m , zp⊗ξl−p+mi,
hκl,m b
⊗pzp, ξl+m−pi=hκl,m, ξl+m−p⊗zpi.
Hence, for Ξ = P∞
i,j=0 Ξi,j (κi,j )∈ L(F∗
θ,Fθ),
D−Ξ := D−
t1· · · D−
tmΞ =
∞
X
i,j=0
(j+m)!
j!Ξi,j (κi,j+mb
⊗m(δt1⊗ · ·· ⊗ δtm))
and
D+D−Ξ := D+
s1· · · D+
slD−Ξ
=
∞
X
i,j=0
(i+l)!(j+m)!
i!j!Ξi,j ((δs1⊗ · ·· ⊗ δsl)b
⊗lκi+l,j+mb
⊗m(δt1⊗ · ·· ⊗ δtm)).(3.8)
Therefore the Wick symbol of the right hand side of (3.3) can be rewritten as
σ(TQ
−c,−dΞ)(ξ, η)
=
∞
X
i,j,l,m=0
(i+l)!(j+m)!
i!j!l!m!ZRl+m
d(s1)· · · d(sl)c(t1)· · · c(tm)
× h(δs1⊗ · ·· ⊗ δsl)b
⊗lκi+l,j+mb
⊗m(δt1⊗ · ·· ⊗ δtm), η⊗i⊗ξ⊗ji
ds1· · · dsldt1· · · dtm(3.9)
=
∞
X
i,j,l,m=0
(l+i)!(m+j)!
i!j!l!m!hd⊗lb
⊗lκi+l,j+mb
⊗mc⊗m, η⊗i⊗ξ⊗ji.(3.10)
A comparison with (3.5) completes the proof.
8

Lemma 3.4 For any x, y, c, d ∈N, we have
TQ
−c,−dΞx,y =σ(Ξx,y)(c, d)Ξx,y.
Proof. Let x, y, c, d ∈N. Then,
T−c,−dσ(Ξx,y)(ξ, η) = σ(Ξx,y )(ξ+c, η +d)
= exp{hx, η +di+hy, ξ +ci}
= exp{hx, di+hy, ci}σ(Ξx,y )(ξ, η).
Therefore, we obtain
TQ
−c,−dΞx,y = exp{hx, di+hy, ci}Ξx,y =σ(Ξx,y )(c, d)Ξx,y
which gives the desired statement.
Let Φ ∈ F ∗
θ. Then the multiplication operator by Φ is defined by
hhMΦf, gii =hhΦ, f.gii, f, g ∈ Fθ.
From Ref. [14], we recall the Fock expansion of MΦ:
MΦ=
∞
X
l,m=0 l+m
mΞl,m(Φl+m).(3.11)
We observe that for Φ = (Φn)n≥0and φ0= (1,0,0,· · · ) we have MΦφ0= Φ. Moreover,
Φ7→ MΦyields a continuous injection from F∗
θinto L(Fθ,F∗
θ) and if ϕ∈ Fθthen
Mϕbelongs to L(Fθ,Fθ). In order to give a comparison with the classical case, we
can extend the same results in the previous sections from the algebra L(F∗
θ,Fθ) to the
algebra L(Fθ,Fθ). More precisely, motivated by the discussions in sections 4 and 5 in
Ref. [1], one can show that TQ
−c,−dand CQ
Sare continuous linear operators acting on
L(Fθ,Fθ).
Theorem 3.5 Let c, d ∈N. Then, we have
(TQ
−c,−dMϕ)φ0=τ−(c+d)ϕ, ϕ ∈ Fθ,
where τ−(c+d)is the classical translation operator acting on test functions.
Proof. Let c, d ∈N. From the definition of TQ
−c,−dwe obtain
σ(TQ
−c,−dMϕ)(ξ, η) = T−c,−dσ(Mϕ)(ξ, η)
=hhϕ, eξ+c+η+diie−hξ+c,η+di
=L(ϕ)(ξ+c+η+d)e−hξ+c,η+di.
Hence we deduce
L(TQ
−c,−dMϕφ0)(η) = hh(TQ
−c,−dMϕ)φ0, eηii
=hh(TQ
−c,−dMϕ)K, φ0⊗eηii
=σ(TQ
−c,−dMϕ)(0, η)ehc,η+di
=L(ϕ)(c+d+η),(3.12)
9

where ΞKis the kernel of the operator Ξ.
By a direct computation, for ϕ∼(ϕn)n≥0in Fθ, we have
τ−yϕ∼ ∞
X
n=0 n+k
ky⊗nb
⊗nϕn+k!k≥0
, y ∈N.
Therefore,
L(τ−yϕ)(η) =
∞
X
n,k=0 n+k
khy⊗nb
⊗nϕn+k, η⊗ki
=
∞
X
n=0
hϕn,
n
X
k=0 n
kη⊗k⊗y⊗n−ki
=
∞
X
n=0
hϕn,(y+η)⊗ni
=L(ϕ)(y+η)
=τ−yL(ϕ)(η).
Choosing y=c+din Eq. (3.12) we obtain the desired result.
4QWN-Convolution Operators
Let S=P∞
l,m=0 Ξl,m(sl,m) in L(Fθ,F∗
θ) and Ξ = P∞
l,m=0 Ξl,m(κl,m) in L(F∗
θ,Fθ),
where κl,m ∈(N⊗(l+m))sym(l,m)and sl,m ∈(N⊗(l+m))0
sym(l,m). Then, the duality between
the spaces L(F∗
θ,Fθ) and L(Fθ,F∗
θ), denoted by hhh·,·iii, is defined as follows
hhhS, Ξiii :=
∞
X
l,m=0
l!m!hsl,m, κl,m i.(4.1)
Proposition 4.1 For S∈ L(Fθ,F∗
θ)and Ξ∈ L(F∗
θ,Fθ), the function Ψdefined on
N×Nby
Ψ(c, d) = hhhS, T Q
−c,−dΞiii
belongs to Hθ(N⊕N). Moreover, for all α≥0,γ1, γ2>0, there exist C1>0,α0≥0
and γ0
1,γ0
2>0such that
kΨkθ,α,(γ1,γ2)≤C1|||Ξ|||θ,α0,(γ0
1,γ0
2)|||S|||θ,−α0,(γ0
1,γ0
2).
Proof. Let S=P∞
i,j=0 Ξi,j (si,j )∈ L(Fθ,F∗
θ) and Ξ = P∞
l,m=0 Ξl,m(κl,m)∈
L(F∗
θ,Fθ). For c,d ∈N, using (3.5), we have
Ψ(c, d) = hhhS, T Q
−c,−dΞiii
=
∞
X
i,j=0
i!j!hsi,j ,
∞
X
l,m=0 i+l
ij+m
jd⊗lb
⊗lκl+i,m+jb
⊗mc⊗mi
=
∞
X
i,j,l,m=0
(l+i)!(m+j)!
l!m!hsi,j b
⊗i
jκl+i,m+j, d⊗l⊗c⊗mi.
10

Then, for α≥0 and γ1,γ2>0, we get
|Ψ(c, d)| ≤
∞
X
i,j,l,m=0 θ−1
l+iθ−1
m+j(γ1
4)−(l+i)(γ2
4)−(m+j)|κl+i,m+j|α
×i!j!θiθjγi
1γj
2|si,j |−αγ1|d|α)lθl(γ2|c|αmθm
≤ k−−→
σ(Ξ)kθ,α,(( γ1
4)2,(γ2
4)2)k−−→
σ(S)kθ,−α,(γ2
1,γ2
2)eθ(γ1|d|α)eθ(γ2|c|α).
Using the same arguments as in Theorem 2 and Theorem 4 in [12] (see also [15]), there
exists a constant C1>0, α0≥0 and γ0
1,γ0
2>0 such that
|Ψ(c, d)| ≤ C1eθ(γ1|d|α)eθ(γ2|c|α)|||Ξ|||θ,α0,(γ0
1,γ0
2)|||S|||θ,−α0,(γ0
1,γ0
2).(4.2)
From which we deduce that Ψ ∈ Hθ(N⊕N).
From Proposition 4.1 and Proposition 2.2, we naturally introduce the notion of con-
volution product in the quantum white noise context.
Definition 4.2 The convolution product of S∈ L(Fθ,F∗
θ)and Ξ∈ L(F∗
θ,Fθ)is defined,
via the Wick symbol transform σ, as the unique element of L(F∗
θ,Fθ), denoted by S ? Ξ,
as
σ(S ? Ξ)(c, d) = hhhS, T Q
−c,−dΞiii, c, d ∈N. (4.3)
Lemma 4.3 For any S∈ L(Fθ,F∗
θ), we have
S ? Ξx,y =hhhS, Ξx,y iii Ξx,y ,∀x, y ∈N.
Proof. Let x, y ∈N. Using Lemma 3.4 we get
σ(S ? Ξx,y )(c, d) = hhhS, T Q
−c,−dΞx,yiii
=σ(Ξx,y)(c, d)hhhS, Ξx,yiii.
That is
S ? Ξx,y =hhhS, Ξx,y iiiΞx,y ,∀x, y ∈N
as desired.
Definition 4.4 AQWN-convolution operator on L(F∗
θ,Fθ)is a linear continuous map-
ping CQwhich commutes with all QWN-translation operators. More precisely, for any
Ξ∈ L(F∗
θ,Fθ)and c,d∈Nwe have
CQ(TQ
−c,−dΞ) = TQ
−c,−d(CQΞ).
The set of all QWN-convolution operators on L(F∗
θ,Fθ) will be denoted CQ.
Theorem 4.5 Let S∈ L(Fθ,F∗
θ)be given. Then, the mapping
CQ
S(Ξ) := S ? Ξ,Ξ∈ L(F∗
θ,Fθ)
is a QWN-convolution operator. Conversely, for any CQ∈CQthere exists a unique
S∈ L(Fθ,F∗
θ)such that CQ=CQ
S.
11

Proof. It is obvious that CQ
Sis a continuous linear mapping. Let us check the
commutation with QWN-translations operators. Given c, d ∈Nit holds that
σ(CQ
S(TQ
−c,−dΞ))(ξ, η) = σ(S ? (TQ
−c,−dΞ))(ξ, η)
=hhhS, T Q
−ξ,−η(TQ
−c,−dΞ)iii
=hhhS, T Q
−ξ−c,−η−dΞiii.
On the other hand,
σ(TQ
−c,−d(CQ
SΞ))(ξ, η) = T−c,−dσ(S ? Ξ)(ξ, η)
=σ(S ? Ξ)(ξ+c, η +d)
=hhhS, T Q
−ξ−c,−η−dΞiii.
This completes the proof of the first statement.
To prove the second statement we define a linear mapping between CQand L(Fθ,F∗
θ)
as follows: for any CQ∈CQwe associate S∈ L(Fθ,F∗
θ) defined by
hhhS, Ξiii := σ(CQΞ)(0,0).
Then it is sufficient to prove that CQ
S=CQ. For c, d ∈Nwe have
σ(CQ
SΞ)(c, d) = σ(S ? Ξ)(c, d)
:= hhhS, T Q
−c,−dΞiii
:= σ(CQ(TQ
−c,−dΞ))(0,0)
=σ(TQ
−c,−d(CQΞ))(0,0)
=T−c,−dσ(CQΞ)(0,0)
=σ(CQΞ)(c, d).
This proves the existence part. The uniqueness follows immediately from Lemma 4.3
and a density argument.
It is shown (see Ref. [12]) that Gθ∗(N⊕N) is closed under pointwise multiplication.
Then, for any S1, S2∈ L(Fθ,F∗
θ), there exists a unique Ξ ∈ L(Fθ,F∗
θ), denoted S1S2,
such that
σ(S1S2) = σ(S1)σ(S2).(4.4)
The operator S1S2will be referred to as the Wick product of S1and S2. It is noteworthy
that, endowed with the Wick product ,L(Fθ,F∗
θ) becomes a commutative algebra. To
each S1∈ L(Fθ,F∗
θ) we associate the QWN-Wick multiplication operator M
S1:
M
S1S2=S1S2, S2∈ L(Fθ,F∗
θ).
Theorem 4.6 Let S1,S2∈ L(Fθ,F∗
θ). Then, we have
(CQ
S1)∗=M
S1and CQ
S1◦CQ
S2=CQ
S1S2.
12

Proof. Let S1,S2∈ L(Fθ,F∗
θ). Then, we get
hhh(CQ
S1)∗S2,Ξx,yiii =hhhS2, S1?Ξx,y iii, x, y ∈N.
Moreover using Lemma 4.3, we have
hhh(CQ
S1)∗S2,Ξx,yiii =hhhS1,Ξx,y iiihhhS2,Ξx,y iii.(4.5)
On the other hand, for S=P∞
l,m=0 Ξl,m(κl,m)∈ L(Fθ,F∗
θ), we have
hhhS, Ξx,yiii =
∞
X
l,m=0
l!m!κl,m,x⊗l
l!⊗y⊗m
m!
=σ(S)(y, x).
Then equation (4.5) becomes
σ((CQ
S1)∗S2)(y, x) = σ(S1)(y, x)σ(S2)(y, x)
=σ(S1S2)(y, x)
=σ(M
S1S2)(y, x).
This proves of the first identity.
Now let S1,S2∈ L(Fθ,F∗
θ). Then, for Ξ ∈ L(F∗
θ,Fθ) we have
σ(CQ
S1(CQ
S2(Ξ)))(c, d) = hhhS1, T Q
−c,−d(CQ
S2Ξ)iii
=hhhS1, CQ
S2(TQ
−c,−dΞ)iii
=hhhS1S2, T Q
−c,−dΞiii
=σ(CQ
S2S2(Ξ))(c, d).
Theorem 4.7 Let S=P∞
i,j=0 Ξi,j (si,j )∈ L(Fθ,F∗
θ). Then, the QWN-convolution opera-
tor CQ
Sadmits the following integral representation
CQ
S=
∞
X
i,j=0 ZRi+j
si,j (u1,· · · , ui, v1,· · · , vj)D+
u1· · · D+
uiD−
v1· · · D−
vjdu1· · · duidv1· · · dvj
(4.6)
on L(F∗
θ,Fθ).
Proof. Let S=P∞
i,j=0 Ξi,j (si,j )∈ L(Fθ,F∗
θ) and Ξ = P∞
l,m=0 Ξl,m(κl,m)∈
L(F∗
θ,Fθ). For c,d∈N, using (3.5), we have
σ(CQ
SΞ)(c, d)
=hhhS, T Q
−c,−dΞiii
=
∞
X
i,j=0
i!j!hsi,j ,
∞
X
l,m=0 l+i
im+j
jd⊗lb
⊗lκl+i,m+jb
⊗mc⊗mi
=
∞
X
i,j,l,m=0
(l+i)!(m+j)!
l!m!hsi,j b
⊗i
jκl+i,m+j, d⊗l⊗c⊗mi.(4.7)
13

On the other hand from (3.8) we get
D+
u1· · · D+
uiD−
v1· · · D−
vjΞ
=
∞
X
l,m=0
(l+i)!(m+j)!
l!m!Ξl,m((δu1⊗ · · · ⊗ δui)b
⊗iκi+l,,j+mb
⊗j(δv1⊗ · ·· ⊗ δvj)).
Denoting the right hand side of (4.6) by AQwe write
σ(AQΞ)(c, d)
=
∞
X
i,j,l,m=0
(l+i)!(m+j)!
l!m!ZRl+m
si,j (u1,· · · , ui, v1,· · · , vj)
h(δu1⊗ · ·· ⊗ δui)b
⊗iκl+i,m+jb
⊗j(δv1⊗ · ·· ⊗ δvj), d⊗l⊗c⊗midu1· · · duidv1· · · dvj
=
∞
X
i,j,l,m=0
(l+i)!(m+j)!
l!m!hsi,j b
⊗i
jκl+i,m+j, d⊗l⊗c⊗mi,
as desired.
Example 4.8 From the fact
D−
zΞx,y =hz, yiΞx,y, D+
zΞx,y =hz, xiΞx,y , x, y, z ∈N
and Lemma 4.3 we get
D−
zΞx,y =CQ
a(z)Ξx,y, D+
zΞx,y =CQ
a∗(z)Ξx,y.
Then by a density argument we have
D−
z=CQ
a(z), D+
z=CQ
a∗(z).
From Theorem 4.6 one observe that for all S∈ L(Fθ,F∗
θ),
(D−
z)∗S=a(z)S, (D+
z)∗S=a∗(z)S.
In conclusion, D±
zare QWN-convolution operators and their adjoint (D±
z)∗are QWN-Wick
multiplication operators.
Example 4.9 The QWN-(K1, K2)-Gross Laplacian ∆Q
G(K1, K2)defined in [2] and repre-
sented by
∆Q
G(K1, K2) = ZR2
τK2(s, t)D−
sD−
tdsdt +ZR2
τK1(s, t)D+
sD+
tdsdt
is a QWN-convolution operator. Moreover,
∆Q
G(K1, K2) = CQ
∆G(K2)+∆∗
G(K1)
i.e., for all Ξ∈ L(F∗
θ,Fθ),
∆Q
G(K1, K2)Ξ = {∆G(K2)+∆∗
G(K1)}?Ξ.
From Theorem 4.6 one also deduce that for all S∈ L(Fθ,F∗
θ),
(∆Q
G(K1, K2))∗S={∆G(K2)+∆∗
G(K1)}  S.
14

Theorem 4.10 Let S∈ L(F∗
θ,F∗
θ). Then for ϕ∈ Fθwe have
CQ
SMϕ=MCψ(ϕ),
where CΨis the classical convolution operator and Ψ∈ F∗
θis such that L(Ψ)(η) =
σ(S)(η, η).
Proof. From Ref. [4], we recall that, for Φ ∼(Φm)min F∗
θ, we have CΦ=
P∞
m=0 DΦmwhere DΦm∈ L(Fθ,Fθ) is given by
DΦm(ϕ)(z) =
∞
X
n=0
(n+m)!
n!hz⊗n,Φmb
⊗mϕn+mi, ϕ ∼(ϕn)n≥0∈ Fθ.
On the other hand, for Ψ ∼(Ψn) in F∗
θ, we have
L(Ψ)(η) = σ(S)(η, η)
=
∞
X
i,j=0
hsi,j , η⊗(i+j)i.
The change of variable n=i+jand a simple identification yield
Ψn=
n
X
j=0
sn−j,j .
Therefore,
L(CΨϕ)(η) =
∞
X
i,j,n=0
(n+i+j)!
n!hsi,j b
⊗i+jϕn+i+j, η⊗ni,
which gives (CΨϕ)∼(Fn)nwith Fnbeing the distribution
Fn=
∞
X
i,j=0
(n+i+j)!
n!si,j b
⊗i+jϕn+i+j.
By means of (3.11), we obtain
σ(MCΨϕ)(ξ, η) =
∞
X
i,j,l,m=0 l+m
mhFl+m, η⊗l⊗ξ⊗mi
=
∞
X
i,j,l,m=0 l+m
m(l+m+i+j)!
(l+m)! hsi,j b
⊗i+jϕl+m+i+j, η⊗l⊗ξ⊗mi.
Finally, in view of (4.7) and (3.11), we obtain
σ(CQ
SMϕ)(ξ, η) =
∞
X
i,j,l,m=0
(l+m+i+j)!
l!m!hsi,j b
⊗i+jϕl+m+i+j, η⊗l⊗ξ⊗mi.
15

Remark 4.11 Since [D−
c, D+
d] = 0 we conclude that
TQ
−c,−d=eD−
ceD+
d=e(D−
c+D+
d)=CQ
ea∗(c)ea(d).
Remark 4.12 Recently in Ref. [8] the authors introduce a convolution for white noise
operators as follows
S1∗GS2=S1S2G, S1, S2, G ∈ L(Fθ,F∗
θ).
It is straightforward that ∗Gis different from Mand CQ. In fact, using Theorem 4.6,
we obtain
S1∗GS2=M
S1G(S2) = (CQ
S1G)∗(S2)
for all S1,S2,G∈ L(Fθ,F∗
θ).
5 Differential equations associated with convolution operators
Let θ1and θ2be two fixed young functions. In the following we assume that there
exists a constant α > 0 such that eθ∗
1(r)−1≤αθ∗
2(r) for rlarge enough.
Consider the following Cauchy problem
∂
∂t Ut=CQ
St(Ut)
U0= Ξ ∈ L(F∗
θ2,Fθ2),(5.1)
where {St}tis a continuous L(Fθ1,F∗
θ1)-process.
Lemma 5.1 [12] Let S∈ L(Fθ,F∗
θ), then the operator eSgiven by
eS:=
∞
X
n=0
Sn
n!
belongs to L(F(eθ∗−1)∗,F∗
(eθ∗−1)∗).
Theorem 5.2 The Cauchy problem (5.1) has a unique solution in L(F∗
θ2,Fθ2)repre-
sented as
Ut=e(Rt
0Ssds)?Ξ.(5.2)
Proof. By Picard’s iteration procedure, we prove that the solution of the problem
(5.1) is the one given in the identity (5.2). Uniqueness follows from the general theory.
Denoting Ut≡U(t), we should apply the iteration to the differential equation
∂U (t)
∂t =f(t, U (t))
where f(t, U (t)) = St? U(t) and initial condition U(0) = Ξ ∈ L(F∗
θ2,Fθ2). Then
U1(t) = U0+Zt
0
f(s, U0)ds
= Ξ + Zt
0
CQ
Ss(U0))ds
= Ξ + Zt
0
(Ss?Ξ)ds
= Ξ + Zt
0
Ssds?Ξ.
16

Next we iterate once more to get the second guess :
U2(t) = U0+Zt
0
f(s, U1(s))ds
= Ξ + Zt
0
CQ
Ss(U1(s))ds
= Ξ + Zt
0Ss?Ξ + Zs
0
(Su?Ξ)duds
= Ξ + Zt
0
Ssds?Ξ + Zt
0Zs
0Ss?Su?Ξduds.
But we know from Theorem 4.6 that
Ss?(Su?Ξ) = CQ
Ss◦CQ
Su(Ξ)
=CQ
SsSuΞ
= (SsSu)?Ξ,
then
U2(t) = Ξ + Zt
0
Ssds?Ξ + Zt
0SsZs
0
Sududs?Ξ.
Hence, by induction for any n≥1 we have
Un(t) = Ξ +
n
X
k=1
Υk(t)?Ξ
with
Υk(t) = Zt
0SunZun
0Sun−1· · ·  Zu2
0
Su1du1· · · dun?Ξ.(5.3)
Fact.
For any k≥1, we have
Υk(t) = 1
k!Zt
0
Suduk.
We prove this identity by induction on k.
•The case k= 2. Set G(t) := Rt
0Sudu; i.e. St=d
dt G(t)≡G0(t). Then
Υ2(t) = Zt
0
(SuG(u))du =Zt
0
(G0(u)G(u))du,
or equivalently, d
dtΥ2(t) = G0(t)G(t).
Applying the Wick symbol transform, we get
d
dtσ(Υ2(t)) = σ(G0(t)) σ(G(t)) = d
dtσ(G(t)) σ(G(t)).
17

Now we integrate this last equality between 0 and t:
σ(Υ2(t)) = 1
2(σ(G(t)))2=1
2(σ(G(t)G(t)))
or equivalently,
Υ2(t) = 1
2(G(t))2=Zt
0
Sudu2.
•Suppose the identity holds for the order k−1≥1. Then, from (5.3) we deduce
Υk(t) = Zt
0SuΥk−1(u)du =Zt
0Su1
(k−1)!Zu
0
Ssds(k−1) du.
With similar arguments as for the previous step, one can easily obtains Υk(t) = 1
k!Rt
0Suduk
as desired.
It then follows that for any n≥1,
Un(t) =
n
X
k=0
1
k!Zt
0
Suduk?Ξ
and hence the exact solution of our Cauchy problem (5.1) is given by
U(t) = lim
n→∞ Un(t) = e(Rt
0Ssds)?Ξ.
Now, from Lemma 5.1, the operator e(Rt
0Ssds)belongs to L(F(eθ∗
1−1)∗,F∗
(eθ∗
1−1)∗). Then,
using the assumption eθ∗
1−1≤αθ∗
2, we get
L(F(eθ∗
1−1)∗,F∗
(eθ∗
1−1)∗)⊂ L(Fθ2,F∗
θ2).
Hence, the operator e(Rt
0Ssds)belongs to L(Fθ2,F∗
θ2) and Utis well defined operator in
L(F∗
θ2,Fθ2).
Example 5.3 For St=a(c) + a∗(d)and c, d ∈N, the operator
Ut=et(a(c)+a∗(d)) ?Ξ (5.4)
is the unique solution of the following Cauchy problem
∂
∂t Ut= (D−
c+D+
d)Ut
U0= Ξ ∈ L(F∗
θ2,Fθ2).
Example 5.4 For St= ∆G(K2)+∆∗
G(K1),
Ut=et(∆G(K2)+∆∗
G(K1)) ?Ξ (5.5)
is the unique solution of the quantum white noise heat equation
∂
∂t Ut= ∆Q
G(K1, K2)Ut
U0= Ξ ∈ L(F∗
θ2,Fθ2).
The operator Pt:= et(∆G(I)+∆∗
G(I)) is referred to as QWN-heat semigroup.
18

Let Jbe the continuous linear operator from Ninto itself defined by
J(ξ1+iξ2) = ξ1−iξ2, ξ1, ξ2∈E.
In the remainder of this paper we put
K1=α1J, K2=α2J, α1, α2∈R.
In the following we study the Cauchy problem
∂
∂t Ut+Ut(D+
d)∗Ut+Ut(D−
c)∗Ut= (∆Q
G(K1, K2))∗Ut+AtUt
U0∈ L(Fθ,F∗
θ),(5.6)
where c, d ∈Nand, for all ξ,η∈N,
σ(At)(ξ, η)∈R,∀t≥0 (5.7)
σ(U0(a(c) + a∗(d))) (ξ, η)≥0.(5.8)
Theorem 5.5 The non-linear equation (5.6) admits the unique solution in L(Fθ,F∗
θ)
given by
σ(Ut)(ξ, η)
=σ(U0)(ξ, η) exp Zt
0hK2ξ, ξi+hK1η, ηi+σ(As)(ξ, η)ds
1 + σ(U0)(ξ, η)hd, ηi+hc, ξiZt
0
exp Zr
0hK2ξ, ξi+hK1η, ηi+σ(As)(ξ, η)dsdr−1
.
Proof. Applying the Wick symbol map to (5.6), we get
∂
∂t σ(Ut)(ξ , η)+(hd, ηi+hc, ξi)σ2(Ut)(ξ, η) = hK2ξ, ξi+hK1η, ηi+σ(At)σ(Ut).(5.9)
Put Vt=1
σ(Ut), Eq. (5.9) is equivalent to
−∂
∂t Vt(ξ , η)+(hd, ηi+hc, ξi) = (hK2ξ, ξi+hK1η, ηi+σ(At)(ξ, η))Vt(ξ , η).(5.10)
The solution of the homogeneous equation is given by
(ξ, η)7→ λexp −Zt
0hK2ξ, ξi+hK1η, ηi+σ(As)(ξ, η)ds,
where λis a constant. Then, the solution of (5.10) can be obtained by the method of
variation of constants :
Vt(ξ, η) = V0(ξ, η) exp −Zt
0hK2ξ, ξi+hK1η, ηi+σ(As)(ξ, η)ds
1 + (hd, ηi+hc, ξi)
V0(ξ, η)Zt
0
exp Zr
0hK2ξ, ξi+hK1η, ηi+σ(As)(ξ, η)dsdr.
19

From this we deduce that
σ(Ut)(ξ, η)
=σ(U0)(ξ, η) exp Zt
0hK2ξ, ξi+hK1η, ηi+σ(As)(ξ, η)ds
1 + σ(U0)(ξ, η)(hd, ηi+hc, ξi)Zt
0
exp Zr
0hK2ξ, ξi+hK1η, ηi+σ(As)(ξ, η)dsdr−1
.
It is easy to see that the function
Θt(ξ, η)
= 1 + σ(U0)(ξ, η)(hd, ηi+hc, ξi)Zt
0
exp Zr
0hK2ξ, ξi+hK1η, ηi+σ(As)dsdr
belongs to the space Gθ∗(N⊕N). Moreover, under the conditions (5.7) and (5.8), for
any ξ, η ∈N, we have Θt(ξ, η)6= 0. Then using the same arguments used in [6] (for the
division results in the space Gθ∗(N)), we get
1
Θ∈ Gθ∗(N⊕N)
which implies σ(Ut)∈ Gθ∗(N⊕N).
Remark 5.6 As a non trivial example of operators satisfying the conditions (5.7) and
(5.8), we take
At=
∞
X
l=0
Ξ2l,0(τ⊗l
β1(t)K1) +
∞
X
m=0
Ξ0,2m(τ⊗m
β2(t)K2),
for every real-valued functions β1,β2and
U0=a(J∗c), c ∈E, d = 0.
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