Quantum Ornstein–Uhlenbeck semigroups

Abstract

Based on nuclear infinite-dimensional algebra of entire functions with a certain exponential growth condition with two variables, we define a class of operators which gives in particular three semigroups acting on continuous linear operators, called the quantum Ornstein-Uhlenbeck (O-U) semigroup, the left quantum O-U semigroup and the right quantum O-U semigroup. Then, we prove that the solution of the Cauchy problem associated with the quantum number operator, the left quantum number operator and the right quantum number operator, respectively, can be expressed in terms of such semigroups. Moreover, probabilistic representations of these solutions are given. Eventually, using a new notion of positive white noise operators, we show that the aforementioned semigroups are Markovian.

Full text

INSTITUTE FOR
QUANTUM STUDIES
CHAPMAN
UNIVERSITY
Quantum Stud.: Math. Found. (2015) 2:159–175
DOI 10.1007/s40509-014-0023-5
REGULAR PAPER
Quantum Ornstein–Uhlenbeck semigroups
Hafedh Rguigui
Received: 4 July 2014 / Accepted: 28 September 2014 / Published online: 8 October 2014
© Chapman University 2014
Abstract Based on nuclear infinite-dimensional algebra of entire functions with a certain exponential growth
condition with two variables, we define a class of operators which gives in particular three semigroups acting
on continuous linear operators, called the quantum Ornstein–Uhlenbeck (O–U) semigroup, the left quantum O–
U semigroup and the right quantum O–U semigroup. Then, we prove that the solution of the Cauchy problem
associated with the quantum number operator, the left quantum number operator and the right quantum number
operator, respectively, can be expressed in terms of such semigroups. Moreover, probabilistic representations of
these solutions are given. Eventually, using a new notion of positive white noise operators, we show that the
aforementioned semigroups are Markovian.
Keywords Space of entire function ·Quantum O–U semigroup ·Quantum number operator ·Cauchy problem ·
Positive operators ·Markovain semigroups
Mathematics Subject Classification 46F25 ·46G20 ·46A32 ·60H15 ·60H40 ·81S25
1 Introduction
Piech [25] introduced the number operator N (Beltrami Laplacian) as infinite-dimensional analog of a finite-
dimensional Laplacian. This infinite-dimensional Laplacian has been extensively studied in [18,20] and the refer-
ences cited therein. In particular, Kuo [18] formulated the number operator as continuous linear operator acting on
the space of test white noise functionals. As applications, Kuo [17] studied the heat equation associated with the
number operator N; this solution is related to the Ornstein–Uhlenbeck (O–U) semigroup. Based on the white noise
theory, Kuo formulated the O–U semigroup as continuous linear operator acting on the space of test white noise
functionals; see [18] and references cited therein. In [7], based on nuclear algebra of entire functions, some results
are extended about operator–parameter transforms involving the O–U semigroup.
In this paper, based on nuclear algebra of entire functions with two variables, three semigroups appear naturally:
the quantum, the left quantum and the right quantum O–U semigroups, respectively. We extend some results
H. Rguigui (B)
Department of Mathematics, High School of Sciences and Technology of Hammam Sousse, University of Sousse,
Rue Lamine Abassi, 4011 Hammam Sousse, Tunisia
e-mail: hafedh.rguigui@yahoo.fr
123

160 H. Rguigui
about these semigroups and their infinitesimal generators called quantum, left quantum and right quantum number
operators, respectively. Moreover, we prove that the solution of the Cauchy problems associated with these operators
can be expressed in terms of the O–U semigroups. Such semigroups are shown to be Markovian.
The paper is organized as follows. In Sect. 2, we briefly recall well-known results on nuclear algebras of entire
holomorphic functions. In Sect. 3, we extend some regularity properties about quantum number operator 
N,left
quantum number operator 
N1, right quantum number operator 
N2and quantum O–U semigroups. In Sect. 4,we
construct semigroups with infinitesimal generator −
N,−
N1and −
N2, respectively. Then, we deduce the solution
of the associated Cauchy problems where its probabilistic representations are given. In Sect. 5, using an adequate
definition of positive operators, we prove that these quantum O–U semigroups are Markovian.
2 Preliminaries
First, we review the basic concepts, notations and some results which will be needed in the present paper. The
development of these and similar results can be found in Refs. [7,11,15,20,21,24].
In mathematics, a nuclear space is a locally convex topological vector space such that for any seminorm p we
can find a larger seminorm q, so that the natural map from Vqto Vpis nuclear. Such spaces preserve many of the
good properties of finite-dimensional vector spaces. As main examples of nuclear spaces we recall the Schwartz
space of smooth functions for which the derivatives of all orders are rapidly decreasing and the space of entire
holomorphic functions on the complex plane with θ−exponential growth. Using a separable Hilbert space and a
positive self-adjoint operator with Hilbert–Schmidt inverse, we can construct a real nuclear space. For i=1,2, let
Hibe a real separable (infinite-dimensional) Hilbert space with inner product ·,· and norm |·|0.Let Ai≥1bea
positive self-adjoint operator in Hiwith Hilbert–Schmidt inverse. Then there exist a sequence of positive numbers
1<λ
i,1≤λi,2≤··· and a complete orthonormal basis of Hi,ei,n∞
n=1⊆Dom(Ai),such that
Aiei,n=λi,nei,n,∞

n=1
λ−2
i,n=

A−1
i

2
HS <∞.
For every p∈R,we define:
|ξ|2
p:= ∞

n=1ξ,ei,n2λ2p
i,n=Ap
iξ2
0,ξ∈Hi.
The fact that, for λ>1, the map p→ λpis increasing implies that:
(i) for p≥0, the space (Xi)p,ofallξ∈Hiwith |ξ|p<∞, is a Hilbert space with norm |·|pand, if p≤q, then
(Xi)q⊆(Xi)p;
(ii) denoting by (Xi)−p,the |·|
−p-completion of Hi(p≥0), if 0 ≤p≤q, then (Xi)−p⊆(Xi)−q.
This construction gives a decreasing chain of Hilbert spaces (Xi)pp∈Rwith natural continuous inclusions iq,p:
(Xi)q→(Xi)p(p≤q). Defining the countably Hilbert nuclear space (see, e.g., [12]):
Xi:= projlim p→∞ (Xi)p∼
=
p≥0
(Xi)p,
the strong dual space X
iof Xiis:
X
i:= indlim p→∞ (Xi)−p∼
=
p≥0
(Xi)−p
and the triple
Xi⊂Hi≡H
i⊂X
i(1)
123

Quantum O–U semigroups 161
is called a real standard triple [20]. For i=1,2, let Nibe the complexification of the real nuclear space Xi.For
p∈N, we denote by (Ni)pthe complexification of (Xi)pand by (Ni)−p,respectively, N
ithe strong dual space of
(Ni)pand Ni. Then, we obtain
Ni=proj lim
p→∞(Ni)pand N
i=ind lim
p→∞(Ni)−p.(2)
The spaces Niand N
iare, respectively, equipped with the projective and inductive limit topology. For all p∈N,
we denote by |.|−pthe norm on (Ni)−pand by ., .the C−bilinear form on N
i×Ni. In the following, Hdenote
by the direct Hilbertian sum of (N1)0and (N2)0, i.e., H=(N1)0⊕(N2)0.Forn∈N, we denote by N
⊗n
i
the n-fold symmetric tensor product on Niequipped with the π−topology and by (Ni)
⊗n
pthe n-fold symmetric
Hilbertian tensor product on (Ni)p. We will preserve the notation |.|pand |.|−pfor the norms on (Ni)
⊗n
pand (Ni)
⊗n
−p,
respectively.
Let θbe a Young function, i.e., it is a continuous, convex and increasing function defined on R+and satisfies
the two conditions: θ(0)=0 and limr→∞
θ(r)
r=∞. Obviously, the conjugate function θ∗of θdefined by
∀x≥0,θ
∗(x):= sup
t≥0
(tx −θ(t)),
is also a Young function. For every n∈N,let
(θ)n=inf
r>0
eθ(r)
rn.(3)
Throughout the paper, we fix a pair of Young function (θ1,θ
2). From now on, we assume that the Young functions
θisatisfy
lim
r→∞
θi(r)
r2<∞.(4)
Note that, if a Young function θsatisfies condition (4), there exist constant numbers αand γsuch that
(θ)n≤α2eγ
nn/2
(5)
and, for r>0 such that rγ<1,
∞

n=0
rnn!(θ)2n<∞.(6)
For a complex Banach space (C,·),letH(C)denotes the space of all entire functions on C, i.e., of all
continuous C-valued functions on Cwhose restrictions to all affine lines of Care entire on C. For each m>0,we
denote by Exp(C,θ,m)the space of all entire functions on Cwith θ−exponential growth of finite type m, i.e.,
Exp(C,θ,m)=f∈H(C);fθ,m:= sup
z∈C|f(z)|e−θ(mz)<∞.
The projective system {Exp((Ni)−p,θ,m);p∈N,m>0}gives the space
Fθ(N
i):= proj lim
p→∞;m↓0Exp((Ni)−p,θ,m). (7)
123

162 H. Rguigui
It is noteworthy that, for each ξ∈Ni, the exponential function
eξ(z):= ez,ξ,z∈N
i,
belongs to Fθ(N
i)and the set of such test functions spans a dense subspace of Fθ(N
i).
For all positive numbers m1,m2>0 and all integers (p1,p2)∈N×N, we define the space of all entire functions
on (N1)−p1⊕(N2)−p2with (θ1,θ
2)−exponential growth by
Exp((N1)−p1⊕(N2)−p2,(θ
1,θ
2), (m1,m2))
={f∈H((N1)−p1⊕(N2)−p2);f(θ1,θ2);(p1,p2);(m1,m2)<∞}
where H((N1)−p1⊕(N2)−p2)is the space of all entire functions on (N1)−p1⊕(N2)−p2and
f(θ1,θ2);(p1,p2);(m1,m2)=sup{| f(z1,z2)|e−θ1(m1|z1|−p1)−θ2(m2|z2|−p2)}
for (z1,z2)∈(N1)−p1⊕(N2)−p2. So, the space of all entire functions on (N1)−p1⊕(N2)−p2with
(θ1,θ
2)−exponential growth of minimal type is naturally defined by
Fθ1,θ2(N
1⊕N
2)=proj lim
p1,p2→∞,m1,m2↓0
Exp((N1)−p1⊕(N2)−p2,(θ
1,θ
2), (m1,m2)). (8)
By definition, ϕ∈Fθ1,θ2(N
1⊕N
2)admits the Taylor expansions:
ϕ(x,y)=∞

n,m=0x⊗n⊗y⊗m,ϕ
n,m,(x,y)∈N
1×N
2(9)
where for all n,m∈N,we have ϕn,m∈N
⊗n
1⊗N
⊗m
2and we used the common symbol ., .for the canonical
C−bilinear form on (N⊗n
1×N⊗m
2)×N⊗n
1×N⊗m
2. So, we identify in the next all test function ϕ∈Fθ1,θ2(N
1⊕N
2)
by their coefficients of its Taylors series expansion at the origin (ϕn,m)n,m∈N. As important example of elements in
Fθ1,θ2(N
1⊕N
2), we define the exponential function as follows. For a fixed (ξ, η) ∈N1×N2,
e(ξ,η)(a,b)=(eξ⊗eη)(a,b)=exp{a,ξ+b,η},(a,b)∈N
1×N
2.
Let ϕ∼(ϕn,m)n≥0in Fθ1,θ2(N
1⊕N
2). Then, from [15] for any p1,p2≥0 and m1,m2>0, there exist q1>p1
and q2>p2such that
|ϕn,m|p1,p2≤en+m(θ1)n(θ2)mmn
1mm
2iq1,p1n
HSiq2,p2m
HS
×ϕ(θ1,θ2);(q1,q2);(m1,m2).(10)
Denoted by F∗
θ1,θ2(N
1⊕N
2)the topological dual of Fθ1,θ2(N
1⊕N
2)called the space of distribution on N
1⊕N
2.
In the particular case where N2={0}, we obtain the following identification
Fθ1,θ2(N
1⊕{0})=Fθ1(N
1)
and therefore
F∗
θ1,θ2(N
1⊕{0})=F∗
θ1(N
1).
So, the space Fθ1,θ2(N
1⊕N
2)can be considered as a generalization of the space Fθ1(N
1)studied in [11].
3 Quantum O–U semigroup and quantum number operator
3.1 Quantum O–U semigroup
Let ϕ(y1,y2)=∞
n,m=0y⊗n
1⊗y⊗m
2,ϕ
n,m∈Fθ1,θ2(N
1⊕N
2).Fors,t≥0, let at=1−exp(−2t)and
bt=exp(−t). Then, we define Os,tby
123

Quantum O–U semigroups 163
Os,tϕ(y1,y2)=X
1×X
2
ϕ(asx1+bsy1,atx2+bty2)dμ1(x1)dμ2(x2),
where μjis the standard Gaussian measure on Xj(forj =1,2)uniquely specified by its characteristic function
e−1
2|ξ|2
0=Xj
eix,ξμj(dx), ξ ∈Xj.
Proposition 1 Let s ,t≥0. Then, the operator Os,tis continuous linear from Fθ1,θ2(N
1⊕N
2)into itself.
Proof Let ϕ∈Fθ1,θ2(N
1⊕N
2). For any p1,p2≥0 and m1,m2>0, there exist p
1,p
2≥0 and m
1,m
2>0 such
that Os,tϕ(y1,y2)
≤X
1×X
2|ϕ(asx1+bsy1,atx2+bty2)|dμ1(x1)dμ2(x2)
≤ϕ(θ1,θ2);(p
1,p
2);(m
1,m
2)X
1
exp θ11
2m1|asx1+bsy1|−p1dμ1(x1)
×X
2
exp θ21
2m2|atx2+bty2|−p2dμ2(x2).
Since, for i=1,2, θiare convex, we have
θi1
2mi|asxi+bsyi|−pi≤1
2θi(mi|as||xi|−pi)+1
2θi(mi|bs||yi|−pi).
Therefore, we obtain Os,tϕ(y1,y2)
≤ϕ(θ1,θ2);(p
1,p
2);(m
1,m
2)exp{θ1(m1|bs||y1|−p1)+θ2(m2|bt||y2|−p2)}
×(X1)−p1
exp{θ1(m1|as||x1|−p1)}dμ1(x1)(X2)−p2
exp{θ2(m2|at||x2|−p2)}dμ2(x2).
Recall that, for pi>1 and i=1,2, (Hi,(Xi)−pi)is an abstract Wiener space. Then, under the condition
limr→∞
θi(r)
r2<∞, the measure μisatisfies the Fernique theorem, i.e., there exist some αi>0 such that
(Xi)−pi
exp{αi|xi|2
−pi}dμi(xi)<∞.(11)
Hence, in view of (11), we obtain
Os,tϕ(y1,y2)exp{−θ1(m1|bs||y1|−p1)−θ2(m2|bt||y2|−p2)}
≤Im1,m2
p1,p2ϕ(θ1,θ2);(p
1,p
2);(m
1,m
2),
where the constant Im1,m2
p1,p2is given by
Im1,m2
p1,p2=(X1)−p1
exp{θ1(m1|as||x1|−p1)}dμ1(x1)
×(X2)−p2
exp{θ2(m2|at||x2|−p2)}dμ2(x2).
123

164 H. Rguigui
This follows that

Os,tϕ
(θ1,θ2);(p1,p2);(m1,m2)≤Im1,m2
p1,p2ϕ(θ1,θ2);(p
1,p
2);(m
1,m
2).
This completes the proof. 
Later on, we need the following Lemma for Taylor expansion.
Lemma 1 Fo r s ,t≥0and n,m∈N, we have X
1×X
2(asx1+bsy1)⊗n⊗(atx2+bty2)⊗mdμ1(x1)dμ2(x2)
=[n/2]

k=0
[m/2]

l=0
n!m!a2k
sa2l
tbn−2k
sbm−2l
t
(n−2k)!(m−2l)!2l+kk!l!(τ⊗k
1
⊗y⊗n−2k
1)⊗(τ ⊗l
2
⊗y⊗m−2l
2),
where τiis the usual trace on Nifor i=1,2.
Proof Using the following equality,
(ax +by)⊗n=
n

k=0
n!
k!(n−k)!(ax)⊗k
⊗(by)⊗n−k,
then, for ξ1∈N1and ξ2∈N2, we easily obtain
X
1×X
2
(asx1+bsy1)⊗n⊗(atx2+bty2)⊗mdμ(x1)dμ(x2), ξ ⊗n
1⊗ξ⊗m
2
=
n

k=0
n!
k!(n−k)!ak
sbn−k
sy⊗n−k
1,ξ⊗n−k
1X
1x⊗k
1,ξ⊗k
1dμ1(x1)
×
m

l=0
m!
l!(m−l)!al
tbm−l
ty⊗m−l
2,ξ⊗m−l
2X
2x⊗l
2,ξ⊗l
2dμ2(x2).
We recall the following identity for the Gaussian white noise measure; see [20],
X
ix⊗k
i,ξ⊗k
idμi(xi)=⎧
⎨
⎩
(2j)!
2jj!|ξi|2
0if k =2j
0if k =2j+1
,
from which we deduce that
X
1×X
2
(asx1+bsy1)⊗n⊗(atx2+bty2)⊗mdμ(x1)dμ(x2), ξ ⊗n
1⊗ξ⊗m
2
=[n/2]

k=0
n!a2k
sbn−2k
sy⊗n−2k
1,ξ⊗n−2k
1
(2k)!(n−2k)!
(2k)!|ξ1|2k
2kk!
×[m/2]

l=0
m!a2l
tbm−2l
ty⊗m−2l
2,ξ⊗m−2l
2
(2l)!(m−2l)!
(2l)!|ξ2|2l
2ll!
=[n/2]

k=0
[m/2]

l=0
n!m!a2k
sa2l
tbn−2k
sbm−2l
t
(n−2k)!(m−2l)!2l+kk!l!
×(τ ⊗k
1
⊗y⊗n−2k
1)⊗(τ ⊗l
2
⊗y⊗m−2l
2), ξ ⊗n
1⊗ξ⊗m
2.
123

Quantum O–U semigroups 165
The above equalities hold for all ξ⊗n
1and ξ⊗m
2with ξ1∈N1and ξ2∈N2; thus, the statement follows by the
polarization identity (see [18,20]). 
Now, we can use Lemma (1) to represent Os,tby Taylor expansion.
Proposition 2 Let s ,t≥0, then for any ϕ∈Fθ1,θ2(N
1⊕N
2)given by ϕ(y1,y2)=∞
n,m=0y⊗n
1⊗y⊗m
2,ϕ
n,m,
we have
(Os,tϕ)(y1,y2)=∞

n,m=0y⊗n
1⊗y⊗m
2,gn,m,
where gn,mis given by
gn,m=bn
sbm
t
n!m!
∞

k,l=0
(n+2k)!(m+2l)!
2l+kk!l!a2k
sa2l
t(τ ⊗k
1⊗τ⊗l
2)
⊗2k,2lϕn+2k,m+2l
and, for ξ1∈N1,ξ2∈N2,
(τ ⊗k
1⊗τ⊗l
2)
⊗2k,2l(ξ⊗n+2k
1⊗ξ⊗m+2l
2)=ξ1,ξ
1kξ2,ξ
2l(ξ⊗n
1⊗ξ⊗m
2).
Proof Consider ϕ(ν1,ν2)(z1,z2)=ν1,ν2
n,m=0z⊗n
1⊗z⊗m
2,ϕ
n,mas an approximating sequence of ϕ∈Fθ1,θ2(N
1⊕
N
2). Then, for any pi∈N,i=1,2 and mi>0,there exist M≥0 such that
ϕ(ν1,ν2)(z1,z2)≤Meθ1(m1|z1|−p1)+θ2(m2|z2|−p2).
Hence, in view of (11), we can apply the Lebesgue dominated convergence theorem to get
Os,tϕ(y1,y2)
=∞

n,m=0X
1×X
2(asx1+bsy1)⊗n⊗(atx2+bty2)⊗m,ϕ
n,mdμ1(x1)dμ2(x2).
Then, by Lemma (1),
Os,tϕ(y1,y2)=∞

n,m=0
[n/2]

k=0
[m/2]

l=0
n!m!a2k
sa2l
tbn−2k
sbm−2l
t
(n−2k)!(m−2l)!2l+kk!l!
×(τ ⊗k
1
⊗y⊗n−2k
1)⊗(τ ⊗l
2
⊗y⊗m−2l
2), ϕn,m.
By changing the order of summation (which can be justified easily), we get
Os,tϕ(y1,y2)=∞

k,l=0;
∞

n=2k;
∞

m=2l
n!m!a2k
sa2l
tbn−2k
sbm−2l
t
(n−2k)!(m−2l)!2l+kk!l!
×y⊗n−2k
1⊗y⊗m−2l
2,(τ⊗k
1⊗τ⊗l
2)
⊗2k,2lϕn,m.
Therefore, we sum over n−2k=jfor j≥0 and m−2l=ifor i≥0 to get
Os,tϕ(y1,y2)
=∞

k,l,j,i=0
(j+2k)!(i+2l)!a2k
sa2l
tbj
sbi
t
j!i!2l+kk!l!y⊗j
1⊗y⊗i
2,(τ⊗k
1⊗τ⊗l
2)
⊗2k,2lϕj+2k,i+2l
=∞

j,i=0y⊗j
1⊗y⊗i
2,∞

k,l=0
(j+2k)!(i+2l)!a2k
sa2l
tbj
sbi
t
j!i!2l+kk!l!(τ ⊗k
1⊗τ⊗l
2)
⊗2k,2lϕj+2k,i+2l.
This proves the desired statement. 
123

166 H. Rguigui
Denoting by L(X,Y)to be the space of continuous linear operators from a nuclear space Xto another nuclear
space Y. From the nuclearity of the spaces Fθi(N
i), we have by Kernel Theorem the following isomorphisms:
L(F∗
θ1(N
1), Fθ2(N
2)) Fθ1(N
1)⊗Fθ2(N
2)Fθ1,θ2(N
1⊕N
2). (12)
So, for every ∈L(F∗
θ1(N
1), Fθ2(N
2)), the associated kernel ∈Fθ1,θ2(N
1⊕N
2)is defined by
ϕ, ψ  = ,ϕ⊗ψ,∀ϕ∈F∗
θ1(N
1), ∀ψ∈F∗
θ2(N
2). (13)
Using the topological isomorphism:
L(F∗
θ1(N
1), Fθ2(N
2)) −→ K=∈Fθ1,θ2(N
1⊕N
2), (14)
we can define the quantum O–U semigroup as follows. For the operator Os,tdefined in this section, we write

Os,t=K−1Os,tK∈LL(F∗
θ1(N
1), Fθ2(N
2)).
The operator
Ot,t, denoted by 
Otfor simplicity, is called the quantum O–U semigroup. The operator 
Os,0is called
the left quantum O–U semigroup and the operator 
O0,tis called the right quantum O–U semigroup.
Recall that the classical O–U semigroup studied in [17,18] is defined by
qtϕ(y)=X
i
ϕ(atx+bty)dμ(x), y∈N
i,ϕ∈Fθ(N
i). (15)
Then, we have the following
Proposition 3 Let s ,t≥0, then we have
Os,t=qs⊗qt,
where qtis the classical O–U semigroup.
Proof We can easily check that
qteξi=exp a2
t
2|ξi|2
0ebtξi,for i=1,2
and
Os,te(ξ1,ξ2)=exp a2
s
2|ξ1|2
0+a2
t
2|ξ2|2
0e(bsξ1,btξ2).(16)
Then, since {e(ξ1,ξ2),ξ
1∈N1,ξ
2∈N2}spans a dense subspace of Fθ1,θ2(N
1⊕N
2), we have the result. 
Theorem 1 Let s ,t≥0, then we have

Os,t() =qsq∗
t,∈L(F∗
θ1(N
1), Fθ2(N
2)),
where q∗
tis the adjoint operator of qt.
Proof Let ∈L(F∗
θ1(N
1), Fθ2(N
2)),φ∈F∗
θ1(N
1)and ϕ∈F∗
θ2(N
2). Then, by Proposition 3,wehave

Os,t()φ, ϕ =  Os,t(K), ϕ ⊗φ
=K, (q∗
sϕ) ⊗(q∗
tφ)
=q∗
tφ,q∗
sϕ
=qsq∗
tφ,ϕ,
which gives the result. 
123

Quantum O–U semigroups 167
3.2 Quantum number operator
Let ϕ(x,y)=∞
n,m=0x⊗n⊗y⊗m,ϕ
n,min Fθ1,θ2(N
1⊕N
2), then we define the three following operators by:
Nϕ(x,y):= ∞

n,m=0;(n,m)=(0,0)
(n+m)x⊗n⊗y⊗m,ϕ
n,m.(17)
N1ϕ(x,y):= ∞

n=1,m=0
nx⊗n⊗y⊗m,ϕ
n,m,(18)
N2ϕ(x,y):= ∞

n=0,m=1
mx⊗n⊗y⊗m,ϕ
n,m.(19)
Proposition 4 N,N1and N2are linear continuous operators from Fθ1,θ2(N
1⊕N
2)into itself.
Proof Let p1,p2≥0. From (17), we deduce that
|Nϕ(x,y)|≤∞

n,m=0;(n,m)=(0,0)
(n+m)|x|n
−p1|y|m
−p2|ϕn,m|p1,p2.
Therefore, using the fact that (n+m)≤2n+mand the inequality (10), for q1>p1,q2>p2and m1,m2>0, we
have
|Nϕ(x,y)|≤ϕ(θ1,θ2);(q1,q2);(m1,m2)
×∞

n,m=0{2m1eiq1,p1HS}n|x|n
−p1(θ1)n{2m2eiq2,p2HS}m|y|m
−p2(θ2)m.
Then, using (3), for m
1,m
2>0, m1,m2>0, q1>p1and q2>p2such that
max 2m1
m
1
eiq1,p1HS,2m2
m
2
eiq2,p2HS<1,
we get
Nϕ(θ1,θ2);(p1,p2);(m
1,m
2)≤ϕ(θ1,θ2);(q1,q2);(m1,m2)cp1,p2,q1,q2
where
cp1,p2,q1,q2=1−(2m1
m
1
eiq1,p1HS)−11−(2m2
m
2
eiq2,p2HS)−1
.
Hence, we prove the continuity of N. Similarly, we complete the proof. 
Recall that the standard number operator on Fθi(N
i)is given by
Nϕ(x)=∞

n=1x⊗n,nϕn,(20)
where ϕ(x)=∞
n=0x⊗n,ϕ
n∈Fθi(N
i).
123

168 H. Rguigui
Remark 1 From (17) and (20), we can easily see that N1,N2and Nhave the following decompositions
N1=N⊗I,N2=I⊗N,N=N⊗I+I⊗N,
respectively.
Definition 1 We define the following operator on L(F∗
θ1(N
1), Fθ2(N
2)) by

N1:= K−1(N1)K,
N2:= K−1(N2)K,
N:= K−1NK =
N1+
N2.
The operator 
N1is called left quantum number operator,
N2is called right quantum number operator and

Nis called quantum number operator.
Proposition 5 For any ∈L(F∗
θ1(N
1), Fθ2(N
2)), we have

N1=N, 
N2=N,
N=N+N.
Proof Let ∈L(F∗
θ1(N
1), Fθ2(N
2)). Then, for any ψ∈F∗
θ1(N
1)and ϕ∈F∗
θ2(N
2), we have
 
N1ψ, ϕ = K−1N1Kψ, ϕ
=N1K, ϕ ⊗ψ
=K, (Nϕ) ⊗ψ
=ψ, Nϕ
=Nψ, ϕ,
which follows that, for any ∈L(F∗
θ1(N
1), Fθ2(N
2)),

N1=N.
Similarly, we get  
N2ψ, ϕ = Nψ, ϕ to obtain 
N2=N. Finally, we get

N=
N1+
N2=N+N.
This completes the proof. 
Note that Definition 1holds true on L(Fθ1(N
1), F∗
θ2(N
2)).
4 Cauchy problem associated with quantum number operator
First, we will construct a semigroup {
Qt,t≥0},{
Qs,0,s≥0}and {
Q0,t,t≥0}on L(F∗
θ1(N
1), Fθ2(N
2))
with infinitesimal generator −
N,−
N1and −
N2, respectively. It reminds constructing a semigroup {Qt,t≥0},
{Qs,0,s≥0}and {Q0,t,t≥0}on Fθ1,θ2(N
1⊕N
2)with infinitesimal generator −N,−N1and −N2,
respectively. Observe that symbolically Qs,t=e−sN1−tN2. Thus, we can define the operator Qs,tas follows. For
ϕ∼(ϕn,m), we define
Qs,tϕ(x,y):= ∞

n,m=0x⊗n⊗y⊗m,e−sn−tmϕn,m,(21)
and let Qt,tdenoted by Qt.
Lemma 2 For any s,t≥0, the linear operator Qs,tis continuous from Fθ1,θ2(N
1⊕N
2)into itself.
123

Quantum O–U semigroups 169
Proof Let ϕ∼(ϕn,m). For any p1,p2≥0, we have
Qs,tϕ(x,y)≤∞

n,m=0
e−sn−tm|x|n
−p1|y|m
−p2|ϕn,m|p1,p2
≤∞

n,m=0|x|n
−p1|y|m
−p2|ϕn,m|p1,p2.
Therefore, using the inequality (10), for q1>p1,q2>p2and m1,m2>0, we get
Qs,tϕ(x,y)≤ϕ(θ1,θ2);(q1,q2);(m1,m2)
×∞

n,m=0{m1eiq1,p1HS}n|x|n
−p1(θ1)n{m2eiq2,p2HS}m|y|m
−p2(θ2)m.(22)
Then, using (3), for m
1,m
2>0, m1,m2>0, q1>p1and q2>p2such that
max m1
m
1
eiq1,p1HS,m2
m
2
eiq2,p2HS<1,
we get
Qs,tϕ(θ1,θ2);(p1,p2);(m
1,m
2)≤ϕ(θ1,θ2);(q1,q2);(m1,m2)Kp1,p2,q1,q2,(23)
where Kp1,p2,q1,q2is given by
Kp1,p2,q1,q2=1−m1
m
1
eiq1,p1HS−11−m2
m
2
eiq2,p2HS−1
.
This proves the desired statement. 
Remark 2 Using (21), Lemma 2, Proposition 2and a similar classical argument used in [18], we can show that
Qs,t=Os,t. Moreover, we see that
Qs,t:= K−1Qs,tK=
Os,t∈L(L(F∗
θ1(N
1), Fθ2(N
2)));
in particular,
Qt=
Ot,
Qs,0=
Os,0and
Q0,t=
O0,t.
Theorem 2 The families {
Qt,t≥0},{
Qs,0,s≥0}and {
Q0,t,t≥0}are strongly continuous semigroup of
continuous linear operators from L(F∗
θ1(N
1), Fθ2(N
2)) into itself with the infinitesimal generator −
N,−
N1
and −
N2,respectively. Moreover, the quantum Cauchy problems
!dt
dt =−
Nt
0=∈L(F∗
θ1(N
1), Fθ2(N
2)) (24)
!ds
ds =−
N1s
0=∈L(F∗
θ1(N
1), Fθ2(N
2)) (25)
!dϒt
dt =−
N2ϒt
ϒ0=∈L(F∗
θ1(N
1), Fθ2(N
2)) (26)
have a unique solutions given respectively by
t=
Qt, s=
Qs,0and ϒt=
Q0,t. (27)
123

170 H. Rguigui
Proof We start by proving that the family {Qt,t≥0}is a strongly continuous semigroup of continuous linear
operators from Fθ1,θ2(N
1⊕N
2)into itself with the infinitesimal generator −Nand the function U(t,x1,x2)=
Qtϕ(x1,x2)satisfies
!∂U(t,x1,x2)
∂t=−NU(t,x1,x2),
limt→0+U(t,x1,x2)=ϕin Fθ1,θ2(N
1⊕N
2).
To this end, it is obvious that QtQs=Qt+sfor any t,s≥0. Thus, we should show the strong continuity of
{Qt,t≥0}. Suppose t≤1, then we can use the inequality |ex−1|≤|x|e|x|,x∈R, to obtain
|Qtϕ(x,y)−ϕ(x,y)|≤∞

n,m=0
(e−t(n+m)−1)|x|n
−p1|y|m
−p2|ϕn,m|p1,p2
≤t∞

n,m=0
e(n+m)|x|n
−p1|y|m
−p2|ϕn,m|p1,p2.
Then, similarly to the proof of Lemma 2, for any q1>p1,q2>p2and m1,m2,m
1,m
2>0 such that
max m1
m
1
e2iq1,p1HS,m2
m
2
e2iq2,p2HS<1,
we get
Qtϕ−ϕ(θ1,θ2);(p1,p2);(m
1,m
2)
≤tϕ(θ1,θ2);(q1,q2);(m1,m2)1−m1
m
1
e2iq1,p1HS1−m2
m
2
e2iq2,p2HS−1
.
This implies the strong continuity of {Qt,t≥0}. To check that −Nis the infinitesimal generator of {Qt,t≥0},
let
Qtϕ−ϕ
t+Nϕ∼(Qn,m),
where Qn,mis given by
Qn,m=!e−t(n+m)+t(n+m)−1
t"ϕn,m,
which follows that, for p1,p2≥0,
Qn,mp1,p2≤
e−t(n+m)−1+t(n+m)
tϕn,mp1,p2.
Using the obvious inequality |ex−1−x|≤x2e|x|for all x∈R, we get
Qn,mp1,p2≤|t|(n+m)2e|t|(n+m)|ϕn,m|p1,p2.
By using (10) and the inequality (n+m)2≤22n+2m, we get, for q1>p1,q2>p2and m1,m2>0,
Qn,mp1,p2≤tϕ(θ1,θ2);(q1,q2);(m1,m2)
×(4m1eiq1,p1HSet)n(4m2eiq2,p2HSet)m(θ1)n(θ2)m.
123

Quantum O–U semigroups 171
Suppose t≤1. Hence, by (3), for m
1,m
2>0, m1,m2>0, q1>p1and q2>p2such that
max 4m1
m
1
e2iq1,p1HS,4m2
m
2
e2iq2,p2HS<1,
we get




Qtϕ−ϕ
t+Nϕ


(θ1,θ2);(p1,p2);(m
1,m
2)≤tc3ϕ(θ1,θ2);(q1,q2);(m1,m2)
where c3is given by
c3=1−4m1
m
1
e2iq1,p1HS−11−4m2
m
2
e2iq2,p2HS−1
.
Then, we obtain
lim
t→0+



Qtϕ−ϕ
t+Nϕ


(θ1,θ2);(p1,p2);(m
1,m
2)=0.(28)
This means that
t−1(Qtϕ−ϕ) −→ −Nϕin Fθ1,θ2(N
1⊕N
2),
i.e., −Nis the infinitesimal generator of {Qt,t≥0}. Moreover, we can write
Qt+sϕ−Qtϕ
s=Qs(Qtϕ) −(Qtϕ)
s.
Since Qtϕ∈Fθ1,θ2(N
1⊕N
2), we can apply (28) to see that the equation
∂U(t,x1,x2)
∂t=−NU(t,x1,x2)
is satisfied by U(t,x1,x2)=Qtϕ(x1,x2). Then, using the topological isomorphism K, we complete the proof of
the first assertion. Similarly, we complete the proof. 
Now, we consider two N
1and N
2-valued stochastic integral equations:
Ut=x+√2t
0
dWs−t
0
Usds
Vt=y+√2t
0
dYs−t
0
Vsds,
where Wtand Ysare standard N
1-valued and N
2-valued Wiener process, respectively, starting at 0.
Theorem 3 The solutions of the Cauchy problems (24), (25)and (26)have the following probabilistic representa-
tions:
K(t)(x,y)=E(f1(Ut)/U0=x)E(f2(Vt)/ V0=y)
K(s)(x,y)=E(g2(y)g1(Us)/U0=x)
K(ϒt)(x,y)=E(h1(x)h2(Vt)/V0=y)
where K(0)=f1⊗f2,K(0)=g1⊗g2,K(ϒ0)=h1⊗h2,f
1,g1,h1∈Fθ1(N
1)and f2,g2,h2∈Fθ2(N
2).
123

172 H. Rguigui
Proof Applying the kernel map Kto the solution (27) of the Cauchy problem (24), we get
K(t)(x,y)=Qt(K(0))(x,y)
=Qt(f1⊗f2)(x,y)
for K(0)=f1⊗f2,f1∈Fθ1(N
1)and f2∈Fθ2(N
2). Then, using Remark 2and Proposition 3, we obtain
K(t)(x,y)=qt(f1)(x)qt(f2)( y).
On the other hand, it is well known from [18] that
qt(f1)(x)=E(f1(Ut)/U0=x), (29)
qt(f2)(y)=E(f2(Vt)/ V0=y), (30)
for f1∈Fθ1(N
1)and f2∈Fθ2(N
2). Similarly, we have
K(s)(x,y)=Qs,0(K(0))(x,y)=Qs,0(g1⊗g2)(x,y)
K(ϒt)(x,y)=Q0,t(K(ϒ0))(x,y)=Q0,t(h1⊗h2)(x,y).
Then, from Proposition 3, we get
K(s)(x,y)=qs(g1)(x)q0(g2)( y)=qs(g1)(x)g2(y)
K(ϒt)(x,y)=q0(h1)(x)qt(h2)( y)=h1(x)qt(h2)(y);
hence, from (29) and (30), we obtain
K(s)(x,y)=E(g1(Us)/U0=x)g2(y)
K(ϒt)(x,y)=h1(x)E(h2(Vt)/V0=y),
which completes the proof. 
5 Markovianity of the quantum O–U semigroups
Recall from [22] that Fθ1,θ2(N
1⊕N
2)is a nuclear algebra with the involution* defined by
ϕ∗(z,w) := ϕ(z,w), z∈N
1,w ∈N
2
for all ϕ∈Fθ1,θ2(N
1⊕N
2). Using the isomorphism K, we can define the involution (denoted by the same
symbol*) on L(F∗
θ1(N
1), Fθ2(N
2)) as follows:
∗:= K−1((K())∗), ∀∈L(F∗
θ1(N
1), Fθ2(N
2))·
Since Fθ1,θ2(N
1⊕N
2)is closed under multiplication, there exists a unique element ϕ∈Fθ1,θ2(N
1⊕N
2), such
that
ϕ=K(1)K(2).
Then by the topology isomorphism K, there exists ∈L(F∗
θ1(N
1), Fθ2(N
2)) such that
K() =K(1)K(2), (31)
which is equivalent to
=K−1(K(1)K(2))·(32)
123

Quantum O–U semigroups 173
Denoted by to be the product between 1and 2,
=12·
Note that from (31) we see that the product is commutative. Now, define the following cones
B:= {∗;∈L(F∗
θ1(N
1), Fθ2(N
2))}·
Elements in B are said to be B-positive operators. Let S,T∈L(F∗
θ1(N
1), Fθ2(N
2)); we say that S≤T,if
T−S∈B. Denoted by I0=K−1#1Fθ1,θ2(N
1⊕N
2)$.
Definition 2 AmapP:L(F∗
θ1(N
1), Fθ2(N
2)) →L(F∗
θ1(N
1), Fθ2(N
2)) is said to be
(i) positive if P(B)⊆B
(ii) Markovian, if it is positive and P() ≤I0whenever =∗and ≤I0.
A one-parameter semigroup {Pt,t≥0}on L(F∗
θ1(N
1), Fθ2(N
2)) is said to be positive (resp. Markovian) provided
Ptis positive (resp. Markovian) for all t≥0.
Theorem 4 The quantum O–U semigroup {
Qt,t≥0}, the right quantum O–U semigroup {
Q0,t,t≥0}and the
left quantum O–U semigroup {
Qs,0,s≥0}are Markovian.
Proof Let ∈B, then there exists S∈L(F∗
θ1(N
1), Fθ2(N
2)) such that
=S∗S·
Then, for all t≥0, z∈N
1and w∈N
2,wehave
K
Qt()(z,w) =Qt(K(K−1(K(S∗)K(S))))(z,w)
=Qt((K(S∗))K(S))(z,w)
=(K(S∗)K(S))(e−tz,e−tw)
=Qt(K(S∗))(z,w)Qt(K(S))(z,w).
Using (32), we get

Qt() =K−1(Qt(K(S∗))Qt(K(S)))
=K−1(K(
Qt(S∗))K(
Qt(S)))
=
Qt(S∗)
Qt(S).
On the other hand, we have
K(
Qt(S∗)) =Qt(K(S∗))·
But we know that
S∗=K−1((K(S))∗)·
Then, we get
K(
Qt(S∗))(z,w) =Qt((K(S))∗)(z,w)
=(K(S))∗(e−tz,e−tw)
=K(S)(e−tz,e−tw)
=(QtK(S))(z,w)
=(QtK(s))∗(z,w).
123

174 H. Rguigui
From this we obtain

Qt(S∗)=K−1((QtK(S))∗)
=K−1((K
Qt(S))∗)
=(
Qt(S))∗.(33)
Hence, we get

Qt() =(
Qt(S))∗Qt(S)·
This proves that 
Qt() ∈Bfor all t≥0,which implies that {
Qt;t≥0}is positive. To complete the proof, let
∈L(F∗
θ1(N
1), Fθ2(N
2)) such that ≤I0and =∗. This gives I0−∈B, which means that there exists
T∈L(F∗
θ1(N
1), Fθ2(N
2)), such that
I0−=T∗T·
This implies that
(1−(K()) =((K(T))∗(K(T). (34)
On the other hand, we have
K(I0−
Qt())(z,w) =1−Qt(K())(z,w)
=1−K()(e−tz,e−tw)
=(1−K())(e−tz,e−tw).
Then, using (34), we get
I0−
Qt() =K−1(Qt((K(T))∗)Qt(K(T)))
=K−1({K
QtK−1(K(T))∗}{K
Qt(T)})
=K−1(K(
Qt(T∗))K(
Qt(T)))
=K−1(K(
Qt(T∗)
Qt(T)))
=
Qt(T∗)
Qt(T).
Then, using (33), we obtain
I−
Qt() =(
Qt(T))∗
Qt(T)·
This means that
I0−
Qt() ∈B,
which is equivalent to say that

Qt() ≤I0,∀t≥0·
This completes the proof of the Markovianity of the quantum O–U semigroup {
Qt,t≥0}. Similarly, we show the
Markovianity of the others semigroups. 
Remark 3 Let 1,2∈L(F∗
θ1(N
1), Fθ2(N
2)), define the following scalar product:
(((1,
2))) := X
1×X
2
K(1)(x,y)K(2)(x,y)dμ1(x)dμ2(y)·
Using Theorem 4,{
Qt;t≥0}is a positive semigroup. Let 1,
2∈B,such that 1,
2= 0. Then, there exist
S,T∈L(F∗
θ1(N
1), Fθ2(N
2)), such that
1=S∗S,
2=T∗T.
123

Quantum O–U semigroups 175
From this one can get
(((1,
Qt(2)))) ≥0.
But it is important to show that: for all 1,
2∈B,
1,
2= 0, there exists t>0 such that
(((1,
Qt(2)))) > 0,
i.e., {
Qt,t≥0}is ergodic, which gives scope for new work.
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