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Mathematical Physics, Analysis and Geometry manuscript No.
(will be inserted by the editor)
Euler’s Theorem For Homogeneous White Noise
Operators
Abdessatar Barhoumi ·Hafedh Rguigui
Received: date / Accepted: date
Abstract In this paper we introduce a new notion of λ−order homogeneous op-
erators on the nuclear algebra of white noise operators. Then, we give their Fock
expansion in terms of quantum white noise (QWN) fields {at, a∗
t;t∈R}. The
quantum extension of the scaling transform enables us to prove Euler’s theorem
in quantum white noise setting.
Keywords QWN-Euler operator, Euler’s Theorem, QWN-scaling operator, Homoge-
neous operator, QWN-derivatives.
Mathematics Subject Classification (2000) 60H40, 46A32, 46F25, 46G20.
1 Introduction and Preliminaries
Let Hbe the real Hilbert space of square integrable functions on Rwith norm
| · |0and E≡ S(R) be the Schwartz space consisting of rapidly decreasing C∞-
functions. Then, the nuclear Gel’fand triple
S(R)⊂L2(R, dx)⊂ S0(R) (1)
can be reconstructed in a standard way (see Ref. [18]) by the harmonic oscillator
A= 1 + t2−d2/dt2and H. The eigenvalues of Aare 2n, n = 1,2,· · · , the
corresponding eigenfunctions {en;n≥1}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
A. Barhoumi
Carthage University, Tunisia, Nabeul Preparatory Engineering Institute, Department of Math-
ematics, Campus Universitaire - Mrezgua - 8000 Nabeul
E-mail: abdessatar.barhoumi@ipein.rnu.tn
H. Rguigui
Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El-Manar, 1060
Tunis, Tunisia
E-mail: hafedh.rguigui@yahoo.fr
2 Abdessatar Barhoumi, Hafedh Rguigui
and we have
E= proj lim
p→∞ Ep, E0= ind lim
p→∞ E−p,
where, for p≥0, Epis the completion of Ewith respect to the norm | · |pand E−p
is 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, we fix a Young
function θsatisfying the condition
lim sup
x→∞
θ(x)
x2<+∞· (2)
Its polar function θ∗is the Young function defined by
θ∗(x) = sup
t≥0
(tx −θ(t)), x ≥0.
For more details , see Refs. [8].
For a complex Banach space (B, k · k), H(B) denotes the space of all entire
functions on Band for m > 0, Exp(B, θ, m) is the Banach space
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 nuclear spaces
Fθ(N0) = proj lim
p→∞;m↓0Exp(N−p, θ, m),Gθ(N) = ind lim
p→∞;m→0Exp(Np, θ, m).
(3)
It is noteworthy that, for each ξ∈N, the exponential function
eξ(z) := ehz,ξi, z ∈N0,
belongs to Fθ(N0) and the set of such test functions spans a dense subspace of
Fθ(N0). In the remainder of this paper we use simply Fθto denote the space
Fθ(N0). The space of continuous linear operators from Fθinto its topological dual
space F∗
θis denoted by L(Fθ,F∗
θ) and assumed to carry the bounded convergence
topology. For z∈N0and ϕ∈ Fθwith Taylor expansions P∞
n=0hx⊗n, fni, 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.(4)
We can check that the limit always exists and a(z)∈ L(Fθ,Fθ). Let a∗(z)∈
L(F∗
θ,F∗
θ) be the dual adjoint of a(z), i.e., for Φ∈ F∗
θand φ∈ Fθ,hha∗(z)Φ, φii =
hhΦ, a(z)φii, where hh·,·ii denotes the standard bilinear form on F∗
θ× Fθ. Similarly,
for ψ∈ Gθ∗(N) with Taylor expansion ψ(ξ) = P∞
n=0hψn, ξ ⊗niwe use the common
notation a(z)ψfor the derivative (4) with z∈N.
The Wick symbol of Ξ∈ L(Fθ,F∗
θ) is by definition [18] a C-valued function
on N×Ndefined by
σ(Ξ)(ξ, η) = hhΞeξ, eηiie−hξ,η i, ξ, η ∈N. (5)
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 (3)
by replacing Npby Np×Np, we have the following characterization theorem for
operator Wick symbols.
Euler’s Theorem For Homogeneous White Noise Operators 3
Theorem 1 (See Ref. [13]) The Wick symbol map σyields a topological isomor-
phism between L(Fθ,F∗
θ)and Gθ∗(N⊕N).
It is a fundamental fact in quantum white noise theory [18] (see, also Ref. [13])
that every white noise operator Ξ∈ L(Fθ,F∗
θ) admits a unique Fock expansion
Ξ=
∞
X
l,m=0
Ξl,m(κl,m ),(6)
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 uniquely specified via the Wick symbol transform by
σ(Ξl,m(κl,m ))(ξ, η) = hκl,m, η⊗l⊗ξ⊗mi, ξ, η ∈N. (7)
For any S1, S2∈ L(Fθ,F∗
θ), there exists a unique Ξ∈ L(Fθ,F∗
θ), denoted S1S2,
such that
σ(S1S2) = σ(S1)σ(S2).(8)
The operator S1S2will be referred to as the Wick product of S1and S2.
Let θnbe given by θn= infr>0eθ(r)/rn, n ∈N. Then, for p∈Nand γ1, γ2>0,
we define the Hilbert space
Fθ,γ1,γ2(Np⊕Np) =
n−→
ϕ= (ϕl,m)∞
l,m=0;ϕl,m ∈(N⊗l
p⊗N⊗m
p)sym(l,m),
∞
X
l,m=0
(θlθm)−2γ−l
1γ−m
2|ϕl,m|2
p<∞o
Put
Fθ(N⊕N) = \
p∈N,γ1>0,γ2>0
Fθ,γ1,γ2(Np⊕Np).
Theorem 2 ([4]) An operator Ξ∈ L(F∗
θ,Fθ)if and only if there exists a unique
(κl,m)l,m ∈Fθ(N⊕N)such that
Ξ=Ξ−τ
∞
X
l,m=0
Ξl,m(κl,m ),(9)
where τis the usual trace on N⊗N, i.e., hτ, ξ ⊗ηi=hξ, η iand
Ξ±τ=
∞
X
k=0
(±1)k
k!Ξk,k(τ⊗k).
Let Uθbe the space of white noise operators given by
Uθ=nΞ=
∞
X
l,m=0
Ξl,m(κl,m ); (κl,m)l,m ∈Fθ(N⊕N)o·
For x, y ∈N, we put κl,m (x, y) = x⊗l
l!⊗y⊗m
m!and Ξx,y := P∞
l,m=0 Ξl,m(κl,m (x, y)).
Then, the set {Ξx,y ;x, y ∈N}spans a dense subspace of Uθ.
4 Abdessatar Barhoumi, Hafedh Rguigui
Theorem 3 ([4]) The map fτdefined by
fτ:L(F∗
θ,Fθ)−→ Uθ, Ξ 7−→ ΞτΞ ,
is a topological isomorphism.
We recall from Ref. [4] the dual pairing: for T=P∞
l,m=0 Ξl,m(Φl,m )∈ Uθand
Ξ=P∞
l,m=0 Ξl,m(κl,m )∈ L(Fθ,F∗
θ), we define
Ξ, T :=
∞
X
l,m=0
l!m!hκl,m, Φl,m i.
For more details see [4], [5], [6],[22], [23] and [24].
In mathematics, a homogeneous function is a function with multiplicative scal-
ing behavior: if the argument is multiplied by a factor, then the result is multiplied
by some power of this factor. More precisely, for f∈L2(Rd) and t∈R\{0}, put
Stf(x) = f(tx), x∈Rd. For a given λ∈R, an element f∈L2(Rd) is said to be
λ−order homogeneous if Stf(x) = tλf(x) for each t∈R\{0}and x∈Rd. It is
well known that fis λ−order homogeneous if and only if it satisfies the so-called
Euler equation
d
X
i=1
xi∂
∂xi
f=λf·(10)
In infinite dimension analysis, an analogue of the Euler operator
d
X
i=1
xi∂
∂xi
was
introduced in [17] as follows
∆E=
∞
X
i=1
(a∗(ei) + a(ei))a(ei) =
∞
X
i=1
h · , eiia(ei).
Moreover, the scaling transformation Stis defined at ϕ(x) = P∞
n=0hx⊗n, ϕni ∈ Fθ
by
Stϕ(x) =
∞
X
n=0 Dx⊗n,tn
n!
∞
X
l=0
(t2−1)l(n+ 2l)!
l!2lτ⊗lb
⊗lϕn+2lE, x ∈N0·
For λ∈R,ϕis said to be λ−order homogeneous if Stϕ=tλϕfor any t∈R\{0}.
It is proved in [20] that ϕis λ−order homogeneous if and only if it satisfies the
Euler equation
∆Eϕ=λϕ·(11)
The main purpose of this paper is the study of the QWN-analogue of (11). We start
by introducing a QWN-Scaling transformation and a QWN-second quantization. These
transformations will be used to introduce the notion of λ−order homogeneous
operators. Then, as a first main result we give their Fock expansions (see Theorem
5). Our second main result is stated in Theorem 7, where we show that a white
noise operator Ξis λ−order homogeneous if and only if it satisfies the following
QWN-Euler equation
∆Q
EΞ=λΞ.
Here ∆Q
Eis the QWN-Euler operator defined in [6].
Euler’s Theorem For Homogeneous White Noise Operators 5
2 Fundamental QWN-Operators
2.1 QWN-Laplacians
From [6], the QWN-Gross Laplacian and QWN-conservation operator can be defined
through Theorem 3 on Uθ, respectively, by
∆Q
G=
∞
X
j=1
D+
ejD+
ej+
∞
X
j=1
D−
ejD−
ej,
NQ=
∞
X
j=1
(D+
ej)∗(Dej)++
∞
X
j=1
(D−
ej)∗D−
ej,
where, for ζ∈N,
D+
ζΞ= [a(ζ), Ξ], D−
ζΞ=−[a∗(ζ), Ξ] (12)
are the creation derivative and annihilation derivative of Ξ, (see [12]).
Lemma 1 For any Ξ=P∞
l,m=0 Ξl,m(κl,m )∈ Uθ, we have
NQΞ=
∞
X
l,m=0
(l+m)Ξl,m(κl,m ).(13)
Proof From [6], we have, for x, y ∈N
σ(NQΞx,y)(ξ , η) = (hx, ηi+hy, ξ i)σ(Ξx,y)(ξ, η).
On the other hand, denoting the right hand side of (13) by AQ, we get
σ(AQΞx,y)(ξ , η)
=
∞
X
l=1
∞
X
m=0
hx, ηihx, ηil−1
(l−1)!
hy, ξim
m!+
∞
X
l=0
∞
X
m=1
hx, ηil
l!hy, ξihy , ξim−1
(m−1)!
= (hx, ηi+hy, ξ i)σ(Ξx,y)(ξ, η ).
Then, by a density argument we complete the proof.
It is noteworthy that the identity (13) holds true for Ξ∈ L(Fθ,F∗
θ).
Proposition 1 Let T∈ L(Fθ,F∗
θ). Then, we have
(∆Q
G)∗T={Ξ2,0(τ) + Ξ0,2(τ)} T(14)
(NQ)∗T=NQT. (15)
6 Abdessatar Barhoumi, Hafedh Rguigui
Proof From [1], for Ξ=P∞
l,m=0 Ξl,m(κl,m )∈ Uθ, we have
∆Q
GΞ=
∞
X
l,m=0
(l+2)(l+1)Ξl,m (τ⊗2κl+2,m)+
∞
X
l,m=0
(m+2)(m+1)Ξl,m (κl,m+2⊗2τ),
(16)
where, for zp∈(N⊗p)0, and ξl+m−p∈N⊗(l+m−p),p≤l+m, the contractions
zp⊗pκl,m and κl,m ⊗pzpare defined by
hzp⊗pκl,m, ξl−p+mi=hκl,m , zp⊗ξl−p+mi,
hκl,m ⊗pzp, ξl+m−pi=hκl,m, ξl+m−p⊗zpi.
Then, for T=P∞
l,m=0 Ξl,m(Φl,m )∈ L(Fθ,F∗
θ), we obtain
T, ∆Q
GΞ
=
∞
X
l,m=0
l!m!(l+ 2)(l+ 1)hΦl,m, τ ⊗2κl+2,mi
+
∞
X
l,m=0
l!m!(m+ 2)(m+ 1)hΦl,m, κl,m+2 ⊗2τi
=
∞
X
l=2
∞
X
m=0
l!m!hτ⊗Φl−2,m, κl,m i+
∞
X
l=0
∞
X
m=2
l!m!hΦl,m−2⊗τ, κl,m i.
Therefore, we get
(∆Q
G)∗T=
∞
X
l=2
∞
X
m=0
Ξl,m(τ⊗Φl−2,m ) +
∞
X
l=0
∞
X
m=2
Ξl,m(Φl,m−2⊗τ),
which yields
σ((∆Q
G)∗T)(ξ, η) =
∞
X
l=2
∞
X
m=0
hτ⊗Φl−2,m, η⊗l⊗ξ⊗mi
+
∞
X
l=0
∞
X
m=2
hΦl,m−2⊗τ, η ⊗l⊗ξ⊗mi
={hη, ηi+hξ , ξi}σ(T)(ξ, η)
=σ(Ξ2,0(τ) + Ξ0,2(τ))(ξ, η)σ(T)(ξ, η).
This gives
(∆Q
G)∗T={Ξ2,0(τ) + Ξ0,2(τ)} T
as desired. (15) follows from (13).
Euler’s Theorem For Homogeneous White Noise Operators 7
2.2 QWN-Second Quantization
We start by clarifying the topology of the nuclear algebra L(Fθ,F∗
θ). From The-
orem 1, we have the topological isomorphism:
L(Fθ,F∗
θ)' Gθ∗(N⊕N) = [
p≥0,γ>0
Exp(Np⊕Np, θ∗, γ).
For p≥0 and γ > 0, let Lθ ,−p,γ (Fθ,F∗
θ) denotes the subspace of all Ξ∈
L(Fθ,F∗
θ) which correspond to elements in Exp(Np⊕Np, θ∗, γ). The topology of
Lθ,−p,γ (Fθ,F∗
θ) is naturally induced from the norm of the Banach space Exp(Np⊕
Np, θ∗, γ) which will be denoted by ||| · |||θ,−p,γ , i.e., for Ξ∈ Lθ,−p,γ(Fθ,F∗
θ),
|||Ξ|||θ,−p,γ =kσΞ kθ∗,−p,γ = sup
ξ,η∈Np
|σ(Ξ)(ξ, η)|e−θ∗(γ|ξ|p)−θ∗(γ|η|p).
For Ξ=P∞
l,m=0 Ξl,m(Φl,m )∈ L(Fθ,F∗
θ) and t∈R, we define the operator ΓQ(t)
by
ΓQ(t)Ξ=
∞
X
l,m=0
Ξl,m(tl+mΦl,m ).(17)
We denote by GL(L(Fθ,F∗
θ)) the group of all linear homeomorphisms from L(Fθ,F∗
θ)
onto itself.
Proposition 2 {ΓQ(et)}t∈Ris a regular one-parameter subgroup of GL(L(Fθ,F∗
θ))
with infinitesimal generator NQ.
Proof The proof of the fact that {ΓQ(et)}t∈Ris a one-parameter subgroup of
GL(L(Fθ,F∗
θ)) is straightforward. Since we have
et(l+m)= 1 + t(l+m) + t2
∞
X
k=0
tk
(k+ 2)!(l+m)(k+2)
then, for Ξ=P∞
l,m=0 Ξl,m(Φl,m )∈ L(Fθ,F∗
θ), one can write
ΓQ(et)Ξ=Ξ+tNQ(Ξ) + t2Λ(t)(Ξ) (18)
where
Λ(t)(Ξ) =
∞
X
l,m=0
Ξl,m(Λl,m (t)Φl,m)
with
Λl,m(t) =
∞
X
k=0
tk
(k+ 2)!(l+m)(k+2) .
Now, for |t| ≤ 1, using a similar computation as in [7], one can show that, there
exist c, r, r0>0 and p, q ≥0 such that
σ(ΓQ(et)Ξ)−σ(Ξ)
t−σ(NQΞ)
θ∗,−p,r0
≤c|t|kσ(Ξ)kθ∗,−q,r ·
It then follows
lim
t→0sup
kσ(Ξ)kθ∗,−q,r ≤1
σ(ΓQ(et)Ξ)−σ(Ξ)
t−σ(NQT)
θ∗,−p,γ
= 0.
This proves the desired statement.
8 Abdessatar Barhoumi, Hafedh Rguigui
2.3 QWN-Scaling Transformation
Motivated by the classical case studied in [17] and [20], we define the QWN-scaling
transformation acting on Ξ=P∞
l,m=0 Ξl,m(κl,m )∈ Uθby
SQ
t(Ξ) :=
∞
X
j,k,l,m=0
tl+m(t2−1)j+k(l+ 2j)!(m+ 2k)!
2j+kj!k!l!m!Ξl,m(τ⊗j⊗2jκl+2j,m+2k⊗2kτ⊗k).(19)
We recall from Ref. [6] and Theorem 3 that the QWN-Fourier-Gauss transform
GQ
K1,K2;B1,B2is a continuous linear operator from Uθinto itself defined by
GQ
K1,K2;B1,B2Ξ=
∞
X
l,m
Ξl,m(gl,m ) (20)
where Ki, Bi∈ L(N0, N 0)∩ L(N, N ), i= 1,2 and gl,m is given by
gl,m =
∞
X
j,k=0
(l+ 2j)!(m+ 2k)!
l!m!j!k!B⊗l
1⊗B⊗m
2τ⊗j
K1⊗2jκl+2j,m+2k⊗2kτ⊗k
K2.
In our setting, we observe that SQ
t=GQ
1
2(t2−1)I, 1
2(t2−1)I;tI,tI .
Theorem 4 Let Ξ=P∞
l,m=0 Ξl,m(κl,m )∈ L(Fθ,F∗
θ). Then, (SQ
t)∗Ξis given by
(SQ
t)∗Ξ=F(t)ΓQ(t)(Ξ),
where F(t)is given by
F(t) =
∞
X
j,k=0
(t2−1)j+k
2j+kj!k!Ξ2j,2k(τ⊗j⊗τ⊗k).(21)
Proof For Ξ=Pl,m=0 Ξl,m(κl,m)∈ L(Fθ,F∗
θ) and T=Pl,m=0 Ξl,m(Φl,m )∈
Uθ, we have
Ξ, S Q
tT
=
∞
X
j,k,l,m=0
l!m!Dκl,m,(t2−1)j+ktl+m(l+ 2j)!(m+ 2k)!
2j+kj!l!τ⊗j⊗2jΦl+2j,m+2k⊗2kτ⊗kE
=
∞
X
p,q=0
p!q!D[p/2]
X
j=0
[q/2]
X
k=0
(t2−1)j+k
2j+kj!k!tp+q−2j−2kτ⊗j⊗κp−2j,q−2k⊗τ⊗k, Φp,q E.
This yields
(SQ
t)∗Ξ=
∞
X
p,q=0
[p/2]
X
j=0
[q/2]
X
k=0
tp+q−2j−2k(t2−1)j+k
2j+kj!k!Ξp,q(τ⊗j⊗κp−2j,q −2k⊗τ⊗k).
(22)
Euler’s Theorem For Homogeneous White Noise Operators 9
On the other hand, we have
σ(F(t)ΓQ(t)(Ξ))(ξ, η)
=σ(F(t))(ξ, η)σ(ΓQ(t)(Ξ))(ξ , η)
=
∞
X
j,k,l,m=0
tl+m(t2−1)j+k
2j+kj!k!Dτ⊗j⊗κl,m ⊗τ⊗k, η⊗l+2j⊗ξ⊗m+2kE
=
∞
X
p,q=0
[p/2]
X
j=0
[q/2]
X
k=0
tp+q−2j−2k(t2−1)j+k
2j+kj!k!Dτ⊗j⊗κp−2j,q−2k⊗τ⊗k, η⊗p⊗ξ⊗qE
or equivalently
F(t)ΓQ(t)(Ξ)
=
∞
X
p,q=0
[p/2]
X
j=0
[q/2]
X
k=0
tp+q−2j−2k(t2−1)j+k
2j+kj!k!Ξp,q(τ⊗j⊗κp−2j,q −2k⊗τ⊗k).
Comparing with (22), the statement follows.
Remark 1 Using (19), for x, y ∈N, we have
SQ
t(Ξx,y) = exp 1
2(t2−1)hx, xi+1
2(t2−1)hy, yiΞtx,ty ·(23)
Then, for all s, t ∈R\{0}, by a density argument, one can verify that
SQ
sSQ
t=SQ
st·(24)
In particular, for all s∈Rand t∈R\{0}, we get
SQ
1+ s
t
SQ
t=SQ
s+t·(25)
3 Euler’s Theorem For Homogeneous Operator
3.1 Homogeneous Operator
Definition 1 Let Ξ∈ Uθand λ∈R. We say that Ξis λ-order homogeneous if
for each t∈R\{0}we have
SQ
t(Ξ) = tλΞ. (26)
This definition is motivated by the classical case studied in [17].
Lemma 2 Let l,m ≥0and κl,m ∈(N⊗l⊗N⊗m)sym(l,m). Then, for Ki,Bi∈
L(N0, N 0)∩ L(N, N ),i= 1,2, we have
GQ
K1,K2;B1,B2Ξl,m(κl,m ) =
[l/2]
X
p=0
[m/2]
X
q=0
Ξl−2p,m−2q(gl−2p,m−2q),(27)
where gl−2p,m−2qis given by
gl−2p,m−2q=
l!m!
(l−2p)!(m−2q)!p!q!B⊗(l−2p)
1⊗B⊗(m−2q)
2τ⊗p
K1⊗2pκl,m ⊗2qτ⊗q
K2(28)
and τKiis the Ki-trace defined by hτKi, z ⊗wi=hKiz , wi.
10 Abdessatar Barhoumi, Hafedh Rguigui
Proof The operator Ξl,m (κl,m) can be rewritten as
Ξl,m(κl,m ) =
∞
X
α,β=0
Ξα,β (fα,β ),
where fα,β is defined by
fα,β =κl,m if (l, m) = (α, β)
0if (l, m)6= (α, β).(29)
Then, by using (20), we get
GQ
K1,K2;B1,B2Ξl,m(κl,m ) =
∞
X
α,β=0
Ξα,β (gα,β ),
with
gα,β =
∞
X
j,k=0
(α+ 2j)!(β+ 2k)!
α!β!j!k!B⊗α
1⊗B⊗β
2τ⊗j
K1⊗2jfα+2j,β+2k⊗2kτ⊗k
K2.
From (29), we observe that gα,β = 0 for α>lor β > m. Thus, we obtain
GQ
K1,K2;B1,B2Ξ=X
0≤α≤lX
0≤β≤m
Ξα,β (gα,β ),
with
gα,β =
∞
X
j,k=0 X
2j=l−αX
2k=m−β
l!m!
α!β!j!k!B⊗α
1⊗B⊗β
2τ⊗j
K1⊗2jκl,m ⊗2kτ⊗k
K2.
Moreover, when l−α= 2p+ 1 or m−β= 2q+ 1, we have gα,β = 0. The case
l−α= 2pand m−β= 2qgives
gα,β =l!m!
α!β!p!q!B⊗α
1⊗B⊗β
2τ⊗p
K1⊗2pκl,m ⊗2qτ⊗q
K2.
Replacing αby l−2pand βby m−2q, we get the desired statement.
The following theorem gives the Fock expansion of the λ−order homogeneous
operator in Uθ.
Theorem 5 Let λ∈Nand Ξ=P∞
l,m=0 Ξl,m(κl,m )∈ Uθ. Then, Ξis λ-order
homogeneous if and only if
Ξ=
λ
X
l=0
[l/2]
X
p=0
[λ−l
2]
X
q=0 ZRλ−2p−2q
Υl−2p,λ−l−2q(s1,· · · , sl−2p, t1,· · · , tλ−l−2q)
a∗
s1· · · a∗
sl−2pat1· · · atλ−l−2qds1· · · dsl−2pdt1· · · dtλ−l−2q,
where Υl−2p,λ−l−2qis given by
Υl−2p,λ−l−2q=
∞
X
j,k=0
(l+ 2j)!(λ−l+ 2k)!(−1)j+k
j!k!p!q!(l−2p)!(λ−l−2q)!2j+k2p+qτ⊗(j+p)⊗2(j+p)κl+2j,λ−l+2k⊗2(k+q)τ⊗(k+q).
Euler’s Theorem For Homogeneous White Noise Operators 11
Proof In the following we set
GQ:= GQ
−1
2I,−1
2I;−iI,−iI .
Motivated by the classical case (see [16]), we can show that GQis a topological
isomorphism from Uθinto itself. Moreover,
(GQ)−1Ξ=GQ
−1
2I,−1
2I;iI,iI Ξ . (30)
For any Ξ=P∞
l,m=0 Ξl,m(κl,m )∈ Uθand t∈R, the technical identity
SQ
t(Ξ) = (GQ)−1ΓQ(t)GQ(Ξ) (31)
holds true. Indeed, by direct computation, we have
GQΞx,y = exp −1
2hx, xi − 1
2hy, yiΞ−ix,−iy , x, y ∈N.
Therefore,
ΓQ(t)GQ(Ξx,y) = exp −1
2hx, xi − 1
2hy, yiΞ−itx,−ity .
Then, we obtain
(GQ)−1ΓQ(t)GQ(Ξx,y) = exp 1
2(t2−1)(hx, xi+hy, yi)Ξtx,ty .
Hence, by (23) we deduce that
SQ
t(Ξx,y) = (GQ)−1ΓQ(t)GQ(Ξx,y ),
which proves (31) by density argument.
In view of (31), equation (26) can be rewritten as follows
ΓQ(t)GQ(Ξ) = tλGQ(Ξ).(32)
Let T=P∞
l,m=0 Ξl,m(Φl,m )∈ Uθbe the unique Fock expansion of the operator
T=GQ(Ξ), where Φl,m is given by
Φl,m =
∞
X
j,k=0
(l+ 2j)!(m+ 2k)!(−i)l+m(−1)j+k
l!m!j!k!2j+kτ⊗j⊗2jΦl+2j,m+2k⊗2kτ⊗k.
Then (32) can be rewritten as
ΓQ(t)T=tλT, (33)
or equivalently
∞
X
l,m=0
tl+mΞl,m(Φl,m ) =
∞
X
l,m=0
tλΞl,m(Φl,m ).
12 Abdessatar Barhoumi, Hafedh Rguigui
From the uniqueness of the Fock expansion, this last equation is satisfied if and
only if λ=l+m. Then, Tsatisfies (33) if and only if
T=
λ
X
l=0
Ξl,λ−l(Φl,λ−l).
Therefore, by Eq. (27), we obtain
Ξ= (GQ)−1T
=
λ
X
l=0
GQ
−1
2I,−1
2I;iI,iI (Ξl,λ−l(Φl,λ−l))
=
λ
X
l=0
[l/2]
X
p=0
[λ−l
2]
X
q=0
Ξl−2p,λ−l−2q(Υl−2p,λ−l−2q)
where Υl−2p,λ−l−2qis given by
Υl−2p,λ−l−2q=l!(λ−l)!iλ
(l−2p)!(λ−l−2p)!p!q!2p+qτ⊗p⊗2pΦl,λ−l⊗2qτ⊗q
and
Φl,λ−l=
∞
X
j,k=0
(l+ 2j)!(λ−l+ 2k)!(−1)λ(−1)j+k
l!(λ−l)!j!k!2j+kτ⊗j⊗2jκl+2j,λ−l+2k⊗2kτ⊗k.
Then, we get
Υl−2p,λ−l−2q=
∞
X
j,k=0
(l+ 2j)!(λ−l+ 2k)!(−1)j+k
j!k!p!q!(l−2p)!(λ−l−2q)!2j+k2p+q
×τ⊗(j+p)⊗2(j+p)κl+2j,λ−l+2k⊗2(k+q)τ⊗(k+p)
as desired.
Example 1 (The 1-order homogeneous operators). For z, w ∈N, the operator
Ξ=a∗(z) + a(w)
is a 1-order homogeneous operator. In particular, for z=w, the multiplication
operator Ξ=Mh·,ziis 1-order homogeneous.
Example 2 (The 2-order homogeneous operators). For κ0,2,κ2,0,κ1,1∈N⊗N,
the operators
Ξ2,0(κ2,0) + Ξ0,0(hτ, κ2,0i),
Ξ0,2(κ0,2) + Ξ0,0(hτ, κ0,2i),
Ξ1,1(κ1,1)
are 2-order homogeneous. Note that if we take κ0,2=κ2,0=τKfor K∈ L(N0, N)
such that hτ, τKi= 0, then the K−Gross Laplacian ∆G(K) = Ξ2,0(τK) and its
dual ∆∗
G(K) = Ξ0,2(τK) are 2-order homogeneous operators. Moreover, for B∈
L(N0, N ), the conservation operator N(B) = Ξ1,1(τB) is 2-order homogeneous
operator.
Euler’s Theorem For Homogeneous White Noise Operators 13
Remark 2 Let λ∈N. Then, using Theorem 3, Ξ∈ L(F∗
θ,Fθ) is λ-order homoge-
neous if and only if
Ξ=Ξ−τΞh
where Ξhis λ-order homogeneous in Uθ.
3.2 Euler’s Theorem For Homogeneous Operator
From Ref. [6], the QWN-Euler operator can be represented, via Theorem 3, as a
continuous linear operator on Uθby
∆Q
E:= ∆Q
G+NQ=
∞
X
j=1
MQ+
h·,ejiD+
ej+
∞
X
j=1
MQ−
h·,ejiD−
ej,
where for z∈N0,
MQ−
h·,zi=σ−1(Mh·,zi⊗I)σ, M Q+
h·,zi=σ−1(I⊗Mh·,zi)σ,
and Mh·,ziis the multiplication operator by h·, zi, see [18].
Theorem 6 Let Ξ∈ Uθand t∈R\{0}. Then for each T∈ L(Fθ,F∗
θ)
lim
s→0T, SQ
t+sΞ−SQ
tΞ
s=1
tT, ∆Q
ESQ
tΞ.
Proof By Theorem 4 and Eq. (25) we have
lim
s→0T, SQ
t+sΞ−SQ
tΞ
s= lim
s→01
snF(1 + s
t)ΓQ(1 + s
t)(T)−To, SQ
tΞ
= lim
s→0F(1 + s
t)1
s(ΓQ(1 + s
t)(T)−T)
+1
s(F(1 + s
t)−I)T, SQ
tΞ.
Since, we have
lim
s→0n(1 + s
t)l+m−1
so=d
ds 1 + s
tl+ms=0 =1
t(l+m),
for T=P∞
l,m=0 Ξl,m(Φl,m ), we get
lim
s→0
ΓQ(1 + s
t)(T)−T
s=
∞
X
l,m=0
1
t(l+m)Ξl,m(Φl,m ) = 1
tNQT. (34)
On the other hand using (21), we have
lim
s→0σ(1
s{F(1 + s
t)−I})(ξ, η)
14 Abdessatar Barhoumi, Hafedh Rguigui
= lim
s→0
1
s(exp "(1 + s
t)2−1
2(hξ, ξi+hη, ηi)#−1)
=1
thξ, ξi+hη, ηi
=1
tσΞ0,2(τ) + Ξ2,0(τ)(ξ, η).
Then, from Eqs. (14) and (15) we get
lim
s→0T, SQ
t+sΞ−SQ
tΞ
s=1
tNQT+1
t(∆Q
G)∗T, SQ
tΞ
=1
tT, ∆Q
ESQ
tΞ.
Which gives the desired statement.
Remark 3 Using Theorem 6, for Ξ∈ Uθ, we have
lim
s→0
SQ
esΞ−Ξ
s= lim
s→0es−1
sSQ
(es−1+1)Ξ−Ξ
es−1=∆Q
EΞ.
This shows that {SQ
et}is a semigroup on Uθwith infinitesimal generator ∆Q
E.
Hence, we deduce that SQ
etU0is the unique solution of the Cauchy problem
∂
∂t Ut=∆Q
EUt, U0∈ Uθ.
Theorem 7 (Euler’s theorem). Let Ξ∈ Uθ. Then Ξis λ−order homogeneous if
and only if it satisfies the following QWN-Euler equation
∆Q
EΞ=λΞ·(35)
Proof If Ξis λ-order homogeneous, then by (26) we have
∆Q
EΞ= lim
t→1
SQ
t(Ξ)−SQ
1(Ξ)
t−1= lim
t→1
tλ−1
t−1Ξ=λΞ.
Conversely, suppose that (35) is satisfied. Let t∈R\{0}. Put G(t) = t−λSQ
t(Ξ).
Then, by (35) and Theorem 6 we get
lim
s→0
G(t+s)−G(t)
s= lim
s→0
1
sn(t+s)−λSQ
t+s(Ξ)−t−λSQ
t(Ξ)o
= lim
s→0
1
sn(t+s)−λ(SQ
t+s(Ξ)−SQ
t(Ξ))o
+ lim
s→0
1
sn(t+s)−λ−t−λoSQ
t(Ξ)
=t−(λ+1)∆Q
E(SQ
t(Ξ)) −λt−(λ+1)SQ
t(Ξ).(36)
Now, let T∈ L(Fθ,F∗
θ). Then, by Theorem 4 and Theorem 6, we have
T, SQ
t(∆Q
EΞ)=F(t)ΓQ(t)(T), ∆Q
EΞ
= lim
s→0F(t)ΓQ(t)(T),SQ
1+sΞ−Ξ
s
= lim
s→0T, SQ
tSQ
1+sΞ−SQ
tΞ
s.
Euler’s Theorem For Homogeneous White Noise Operators 15
But, by applying the QWN-Scaling transformation on Ξx,y, for x, y ∈N, we can
show that SQ
uSQ
v=SQ
vSQ
ufor all u, v ∈R. Then we get
T, SQ
t(∆Q
EΞ)= lim
s→0T, SQ
1+sSQ
tΞ−SQ
tΞ
s.
Hence, using Theorem 6 , we obtain
T, SQ
t(∆Q
EΞ)=T, ∆Q
E(SQ
tΞ),
from which we deduce that
SQ
t(∆Q
EΞ) = ∆Q
E(SQ
tΞ).
Therefore, using (35) we get
∆Q
E(SQ
t(Ξ)) = λSQ
t(Ξ).
Thus, from (36) we deduce that G0(t) = 0 for all t∈R\{0}. In particular, G(t) =
G(1), i.e.,
t−λSQ
t(Ξ) = SQ
1(Ξ) = Ξ·
From which we deduce the desired statement.
Remark 4 Euler’s theorem remains valid in L(F∗
θ,Fθ) where ∆Q
Eis replaced by
e
∆Q
Eacting on L(F∗
θ,Fθ) as follows
e
∆Q
E(Ξ) = Ξ−τ∆Q
E(ΞτΞ).
Corollary 1 Let λ∈Nand Ξ=P∞
l,m=0 Ξl,m(κl,m )∈ Uθsuch that ∆Q
G(Ξ)=0.
Then Ξis λ−order homogeneous if and only if Ξ=Pλ
l=0 Ξl,λ−l(κl,λ−l).
Proposition 3 Let Ξ∈ Uθbe a λ−order homogeneous operator such that ∆Q
G(Ξ) =
0. Then for each ξ∈N,D±
ξ(Ξ)are (λ−1)−order homogeneous and (D±
ξ)∗(Ξ)
are (λ+ 1)−order homogeneous.
Proof We recall from [6] that, for any ξ∈N, the following identities hold true
D+
ξΞl,m(κl,m ) = lΞl−1,m(ξ⊗1κl,m )
D−
ξΞl,m(κl,m ) = mΞl,m−1(κl,m ⊗1ξ)
(D+
ξ)∗Ξl,m(κl,m ) = Ξl+1,m(ξ⊗κl,m )
(D−
ξ)∗Ξl,m(κl,m ) = Ξl,m+1(κl,m ⊗ξ).
Then, if Ξ=Pλ
l=0 Ξl,λ−l(κl,λ−l), we have
D+
ξΞ=
λ−1
X
l=0
(l+ 1)Ξl,λ−l−1(ξ⊗1κl+1,λ−1−l).(37)
Thus, the fact ∆Q
G(D+
ξΞ) = D+
ξ(∆Q
GΞ) = 0 and identity (37) proves the state-
ment for D+
ξ(Ξ) via Corollary 1. The others statements can be verified by slight
modification.
16 Abdessatar Barhoumi, Hafedh Rguigui
Theorem 8 Let Ξl,m(κl,m)∈ Uθ. Then Ξl,m (κl,m)is (l+m)−order homogeneous
if and only if ∆Q
G(Ξl,m(κl,m )) = 0.
Proof From (16) we have
∆Q
GΞl,m(κl,m ) = l(l−1)Ξl−2,m(τ⊗2κl,m ) + m(m−1)Ξl,m−2(κl,m ⊗2τ).(38)
Then by iterating (38) we get
(∆Q
G)kΞl,m(κl,m ) =
[l/2]
X
p=0
[m/2]
X
p+q=k
l!m!k!
(l−2p)!(m−2q)!p!q!Ξl−2p,m−2q(τ⊗p⊗2pκl,m ⊗2qτ⊗q).(39)
On the other hand from (27), we obtain
SQ
tΞl,m(κl,m ) =
[l/2]
X
p=0
[m/2]
X
q=0
l!m!
(l−2p)!(m−2q)!p!q!tl+m−2p−2q1
2(t2−1)p+q
×Ξl−2p,m−2q(τ⊗p⊗2pκl,m ⊗2qτ⊗q).(40)
Then in view of (39), (40) becomes
SQ
tΞl,m(κl,m )
=
[(l+m)/2]
X
k=0
1
k!tl+m−2k1
2(t2−1)k(∆Q
G)kΞl,m(κl,m )
=tl+mΞl,m(κl,m ) +
[(l+m)/2]
X
k=1
1
k!tl+m−2k1
2(t2−1)k(∆Q
G)kΞl,m(κl,m ).
It then follows that Ξl,m(κl,m) is (l+m)−homogeneous if and only if
[(l+m)/2]
X
k=1
1
k!tl+m−2k1
2(t2−1)k(∆Q
G)kΞl,m(κl,m ) = 0.(41)
Hence, using the fact that {Pk(X) = Xl+m−2k(X2−1)k;k= 1,2,· · · ,[(l+m)/2]}
is a linearly independent family of polynomials, one can show that (41) holds if
and only if
(∆Q
G)kΞl,m(κl,m ) = 0,∀k= 1,2,· · · ,[(l+m)/2].
This implies in particular that ∆Q
GΞl,m(κl,m ) = 0. The converse is straightforward
by Euler’s theorem.
Euler’s Theorem For Homogeneous White Noise Operators 17
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