Abstract
we develop an operator theory on a nuclear algebra of white noise operators in terms of the quantum white noise (QWN) derivatives and their dual adjoints. Using an adequate definition of a QWN-symbol transformation, we discuss QWN-integral-sum kernel operators which give the Fock expansion of the QWN-operators (i.e. the linear operators acting on nuclear algebra of white noise operators). As application, we characterize all rotation invariant QWN-operators by means of the QWN-conservation operator, the QWN-Gross Laplacians. These topics are expected to open a new area in QWN infinite-dimensional analysis.
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1 Introduction
The white noise analysis has been developed to an infinite-dimensional distribution theory on Gaussian space \((E',\mu )\) as an infinite-dimensional analogue of Schwartz distribution theory on Euclidean space \(\mathbb {R}\) with Lebesgue measure:
The mathematical framework of white noise analysis is the Gel’fand triple of test function space \(\fancyscript{F}_{\theta }(N')\) and generalized function space \(\fancyscript{F}^{*}_{\theta }(N')\):
There has been observed formal analogy between white noise calculus and the calculus on Euclidean space based on this Gel’fand triple, e.g., rotation groups [14], Laplacians [8, 13]. The main tools of investigation in the above-mentioned papers are the symbol (or Wick symbol) transform of an operator, the Fock expansion and integral kernels operators.
In white noise analysis, the set \(\{x(t);\; t \in \mathbb {R} \}\) is taken as a coordinate system of \((E',\mu )\) and \(\{a_{t},\, a_{t}^{*};\; t \in \mathbb {R}\}\) (annihilation and creation) is the coordinate system for white noise differential operators as homologue of the Euclidean differential basis \(\Big \{ \frac{\partial }{\partial x_{k}}\, ;\, 1\le k \le \mathrm{d} \Big \}\). It is a fundamental fact that every white noise operator \(\Xi \in \fancyscript{L}(\fancyscript{F}_{\theta }(N'), \fancyscript{F}^{*}_{\theta }(N'))\) admits a Fock expansion as an infinite series:
where the integral kernel operator \(\Xi _{l,m}(\kappa _{l,m})\) is expressed in a formal integral
Accordingly, the white noise operator \(\Xi \) can be regarded as a “function” of the variables \(\{a_{s},\, a_{t}^{*};\; s,\, t \in \mathbb {R}\}\). This intuitive idea motivated Ji–Obata (see Ref. [16]) to introduce the so-called quantum white noise derivatives
acting on a suitable subset of the nuclear algebra \(\fancyscript{L}(\fancyscript{F}_{\theta }(N'), \fancyscript{F}^{*}_{\theta }(N'))\). The set
will be taken as a quantum white noise coordinate system.
The main purpose of this paper is to develop operator theory on a nuclear algebra of white noise operators; we give the Fock expansion of the QWN-operators in terms of a QWN-integral kernel operators which are defined by the QWN-symbol map and expressed in terms of the quantum white noise coordinate system. The above Fock expansion will play a key role in our discussion, in particular using the quantum white noise analogues of the Gross Laplacian and the number operator, we characterize the rotation-invariant QWN operators.
The paper is organized as follows. In Sect. 2, we summarize the common notations, concepts and basic topological isomorphisms used throughout the paper. In Sect. 3, we introduce the QWN-integral kernel operator using the quantum white noise coordinate system. In Sect. 4, we define the QWN-symbol map and study its properties. In Sect. 5, the chaotic expansion of the QWN-operators is given in terms of the QWN coordinate system. In Sect. 6, we characterize all rotation-invariant QWN-operators.
2 Preliminaries
In this section, we summarize the common notations and concepts used throughout the paper which can be found in Refs. [5–7, 10, 19, 21, 23, 24, 28, 29].
2.1 Basic Gel’fand triples
Let \(H\) be the real Hilbert space of square integrable functions on \(\mathbb {R}\) with norm \(|\cdot |_{0}\). The Gel’fand triple (1) can be reconstructed in a standard way (see Ref. [21]) by the harmonic oscillator \(A=1+t^{2}-d^{2}/dt^{2}\) and \(H\). The eigenvalues of \(A\) are \(2n + 2,\, n = 0, 1, 2, \cdots \) and the corresponding eigenfunctions \(\{e_{n};\; n\ge 0\}\) form an orthonormal basis for \(L^{2}(\mathbb {R})\). In fact, (\(e_{n}\)) are the Hermite functions and therefore each \(e_{n}\) is an element of \(E\). The space \(E\) is a nuclear space equipped with the Hilbertian norms
and we have
where for \(p\ge 0\), \(E_{p}\) is the completion of \(E\) with respect to the norm \(| \cdot |_{p}\) and \(E_{-p}\) is the topological dual space of \(E_{p}\). We denote by \(N=E+iE\) and \(N_{p}=E_{p}+iE_{p}\), \(p\in \mathbb {Z}\), the complexifications of \(E\) and \(E_{p}\), respectively.
Throughout the paper, we fix a Young function \(\theta \) that satisfies the following condition
The polar function \(\theta ^{*}\) of \(\theta \), defined by \(\theta ^{*}(x)=\sup _{t\ge 0}(tx-\theta (t))\), \(x\ge 0\), is also a Young function. For more details, see Refs. [10, 23]. For a complex Banach space \((B, \Vert \cdot \Vert )\), let \(\fancyscript{H}(B)\) denotes the space of all entire functions on \(B\). For each \(m>0\) we denote by \(\text {Exp}(B,\theta ,m)\) to be
The two spaces \(\fancyscript{F}_{\theta }(N')\) and \(\fancyscript{G}_{\theta }(N)\) are defined by
In the remainder of this paper, we simply use \(\fancyscript{F}_{\theta }\) for \(\fancyscript{F}_{\theta }(N')\). It is noteworthy that, for each \(\xi \in N\), the exponential function \(e_{\xi }(z):=e^{\langle z, \xi \rangle }, \quad z\in N'\), belongs to \(\fancyscript{F}_{\theta }\) and the set of such test functions spans a dense subspace of \(\fancyscript{F}_{\theta }\). The space of linear continuous operators from \(\fancyscript{F}_{\theta }\) into its topological dual space \(\fancyscript{F}^{*}_{\theta }\) is denoted by \(\fancyscript{L}(\fancyscript{F}_{\theta }, \fancyscript{F}^{*}_{\theta })\) and assumed to carry the bounded convergence topology. Let \(\mu \) be the standard Gaussian measure on \(E'\) uniquely specified by its characteristic function
Under condition (5), we have the nuclear Gel’fand triple (2), see Ref. [10].
2.2 QWN-derivatives
For \(z \in N'\) and \(\varphi (x)\) with Taylor expansion \(\sum _{n=0}^{\infty }\langle x^{\otimes n},f_{n} \rangle \) in \( \fancyscript{F}_{\theta }\), the holomorphic derivative of \(\varphi \) at \(x \in N'\) in the direction \(z\) is defined by \((a(z)\varphi )(x):=\lim _{\lambda \rightarrow 0}\frac{\varphi (x+\lambda z)-\varphi (x)}{\lambda }\). We can check that the limit always exists, \(a(z) \in \fancyscript{L}(\fancyscript{F}_{\theta }, \fancyscript{F}_{\theta })\) and \(a^{*}(z) \in \fancyscript{L}(\fancyscript{F}^{*}_{\theta }, \fancyscript{F}^{*}_{\theta })\), where \(a^{*}(z)\) is the adjoint of \(a(z)\). For \(\zeta \in N\), \(a(\zeta )\) extends to a continuous linear operator from \(\fancyscript{F}^{*}_{\theta }\) into itself (denoted by the same symbol) and \(a^{*}(\zeta )\) (restricted to \(\fancyscript{F}_{\theta })\) is a continuous linear operator from \(\fancyscript{F}_{\theta }\) into itself. If \(z=\delta _{t}\in E'\) we simply write \(a_{t}\) instead of \(a(\delta _{t})\). In QWN-field theory \(a_{t}\) and \(a_{t}^{*}\) are called the annihilation and creation operators at the point \(t \in \mathbb {R}\).
The symbol and the Wick symbol, denoted by \(\sigma \) and \(\omega \) respectively, of \(\Xi \in \fancyscript{L}(\fancyscript{F}_{\theta }, \fancyscript{F}^{*}_{\theta })\) are by definition ([21]) the \(\mathbb {C}\)-valued function on \(N\times N\) obtained as
respectively, where \(\langle \!\langle \cdot ,\cdot \rangle \!\rangle \) denotes the duality between the two spaces \(\fancyscript{F}^{*}_{\theta }\) and \(\fancyscript{F}_{\theta }\).
It is a fundamental fact in quantum white noise theory [21] (see also Ref. [17]) that every white noise operator \(\Xi \in \fancyscript{L}(\fancyscript{F}_{\theta },\fancyscript{F}^{*}_{\theta })\) admits a unique Fock expansion (3) where, for each pairing \(l,m\ge 0\), \(\kappa _{l,m}\in (N^{\otimes (l+m)})'_{sym (l,m)}\) and \(\Xi _{l,m}(\kappa _{l,m})\) is the integral kernel operator characterized via the Wick symbol transform by
This can be formally rewritten as (4). For \(\zeta \in N\), the quantum white noise derivatives are defined by
These are called the creation derivative and annihilation derivative of \(\Xi \), respectively. In Ref. [1], for \(z\in N'\), it is shown that \(D_{z}^{+}\) is a continuous operator from \(\fancyscript{L}(\fancyscript{F}_{\theta },\fancyscript{F}_{\theta })\) into itself and \(D_{z}^{-}\) is a continuous operator from \(\fancyscript{L}(\fancyscript{F}^{*}_{\theta },\fancyscript{F}^{*}_{\theta })\) into itself. The pointwisely quantum white noise derivatives \(D^{\pm }_{t}\equiv D^{\pm }_{\delta _{t}}\) are discussed in Ref. [16].
2.3 Basic topological isomorphisms
Let \(\fancyscript{G}_{\theta ^{*}}(N \oplus N)\) denotes the nuclear space obtained as in (6) by replacing \(N_{p}\) with \(N_{p}\oplus N_{p}\).
Theorem 1
(See Ref. [17]) The symbol and the Wick symbol maps realize two topological isomorphisms between \(\fancyscript{L}(\fancyscript{F}_{\theta },\fancyscript{F}^{*}_{\theta })\) and \(\fancyscript{G}_{\theta ^{*}}(N \oplus N)\).
For \(p\in \mathbb {N}\) and \(\gamma _1,\gamma _2>0\), we define the Hilbert spaces
where \(\Vert \overrightarrow{\varphi }\Vert ^{2}_{\theta ,p,(\gamma _{1},\gamma _{2})}:=\sum _{l,m= 0}^{\infty } (\theta _l\theta _m)^{-2} \gamma _1^{-l}\gamma _2^{-m}|\varphi _{l,m}|_p^2\) and \(\theta _n=\inf _{r>0}\, e^{\theta (r)}/r^n, \> n\in \mathbb {N}\). Put \(F_\theta (N\oplus N) = \bigcap _{p\in \mathbb {N}, \gamma _1,\gamma _2>0} F_{\theta , \gamma _1,\gamma _2}(N_p\oplus N_p)\). Let \(\fancyscript{H}_{\theta }(N\oplus N)=\bigcap _{p\ge 0,\gamma _{1},\gamma _{2}>0}\text {Exp}(N_{p}\oplus N_{p}, \theta ,\gamma _{1},\gamma _{2})\), where \(\text {Exp}(N_{p}\oplus N_{p}, \theta ,\gamma _{1},\gamma _{2})\) denotes the space of all entire functions on \(N_{p}\times N_{p}\) such that
In other words, from the topological isomorphism between \(\fancyscript{H}_{\theta }(N\oplus N)\) and \(F_{\theta }(N\oplus N)\) which can be easily shown (see [10, 17]), all holomorphic functions \(g\) in \(\fancyscript{H}_{\theta }(N\oplus N)\) admit a Taylor expansion \(g(x_{1},x_{2})=\sum _{l,m}\langle g_{l,m}, x_{2}^{\otimes l}\otimes x_{1}^{\otimes m}\rangle \) for \(x_{1}, x_{2}\in N\), where \(g_{l,m}\in (N^{\otimes (l+m)})_{sym (l,m)}\) is such that \(\Vert \overrightarrow{g}\Vert ^{2}_{\theta ,p,(\gamma _{1},\gamma _{2})}<\infty \) for all \(p\in \mathbb {N}\) and \(\gamma _{1}\), \(\gamma _{2} >0\).
Theorem 2
([5]) The symbol map realizes a topological isomorphism between the space \(\fancyscript{L}(\fancyscript{F}^{*}_{\theta },\fancyscript{F}_{\theta })\) and the space \(\fancyscript{H}_{\theta }(N\oplus N)\).
For any \(S_{1},\, S_{2}\in \fancyscript{L}(\fancyscript{F}_{\theta },\fancyscript{F}^{*}_{\theta })\), the the Wick product of \(S_{1}\) and \(S_{2}\) is defined by
Theorem 3
([5]) An operator \(\Xi \in \fancyscript{L}(\fancyscript{F}_{\theta }^{*}, \fancyscript{F}_{\theta })\) iff there exists a unique \((\kappa _{l,m})_{l,m}\in F_{\theta }(N\oplus N)\) such that
where \(\Xi _{-\tau }\) is given by \( \Xi _{-\tau }=\sum _{k=0}^{\infty }\frac{(-1)^{k}}{k!}\Xi _{k,k}(\tau ^{\otimes k})\).
Let \(\fancyscript{U}_{\theta }=\omega ^{-1}(\fancyscript{H}_{\theta }(N\oplus N))\). By a simple computation one can show that \(\Xi \in \fancyscript{U}_{\theta }\) iff there exists a unique \((\kappa _{l,m})^\infty _{l,m=0}\in F_{\theta }(N\oplus N)\) such that \(\Xi =\sum _{l,m=0}^{\infty }\Xi _{l,m}(\kappa _{l,m})\), i.e.,
Theorem 4
([5]) The Wick symbol map realizes a topological isomorphism between the space \(\fancyscript{U}_{\theta }\) and the space \(\fancyscript{H}_{\theta }(N\oplus N)\).
From Theorems 1 and 2, we have the topological isomorphism:
For \(p\ge 0\) and \(\gamma _{1},\gamma _{2}>0\), let \(\fancyscript{L}_{\theta ,-p,(\gamma _{1},\gamma _{2})}(\fancyscript{F}_{\theta },\fancyscript{F}^{*}_{\theta })\) denotes the subspace of all \(\Xi \in \fancyscript{L}(\fancyscript{F}_{\theta },\fancyscript{F}^{*}_{\theta })\) which correspond to elements in \(\text {Exp}(N_{p}\oplus N_{p}, \theta ^{*}, (\gamma _{1},\gamma _{2}))\). Similarly, let \(\fancyscript{L}_{\theta ,p,(\gamma _{1},\gamma _{2})}(\fancyscript{F}_{\theta }^{*},\fancyscript{F}_{\theta })\) denotes the subspace of all \(\Xi \in \fancyscript{L}(\fancyscript{F}^{*}_{\theta },\fancyscript{F}_{\theta })\) which correspond to elements in \(\text {Exp}(N_{p}\oplus N_{p}, \theta , (\gamma _{1},\gamma _{2}))\). The topology of \(\fancyscript{L}_{\theta ,-p,(\gamma _{1},\gamma _{2})}(\fancyscript{F}_{\theta },\fancyscript{F}^{*}_{\theta })\) is naturally induced from the norm of the Banach space \(\text {Exp}(N_{p}\oplus N_{p}, \theta ^{*}, (\gamma _{1},\gamma _{2}))\) which will be denoted by \(|\!|\!| \cdot |\!|\!|_{\theta ,-p,(\gamma _{1},\gamma _{2})}\), i.e., for \(\Xi \in \fancyscript{L}_{\theta ,-p,(\gamma _{1},\gamma _{2})}(\fancyscript{F}_{\theta },\fancyscript{F}^{*}_{\theta })\),
Similarly, the topology of \(\fancyscript{L}_{\theta ,p,(\gamma _{1},\gamma _{2})}(\fancyscript{F}^{*}_{\theta },\fancyscript{F}_{\theta })\) is naturally induced from the norm of the Banach space \(\text {Exp}(N_{p}\oplus N_{p},\theta ,(\gamma _{1},\gamma _{2}))\) which will be denoted by \(|\!|\!| \cdot |\!|\!|_{\theta ,p,(\gamma _{1},\gamma _{2})}\), i.e., for \(\Xi \in \fancyscript{L}_{\theta ,p,(\gamma _{1},\gamma _{2})}(\fancyscript{F}^{*}_{\theta },\fancyscript{F}_{\theta })\) and for all \(p\ge 0\) and \(\gamma _{1},\gamma _{2}>0\)
Via Theorem 4, the topology of \(\fancyscript{U}_{\theta }\) is governed by the family of seminorms
Theorem 5
([5]) The map \(f_{\tau }\) defined by \(f_{\tau }: \fancyscript{L}(\fancyscript{F}^{*}_{\theta },\fancyscript{F}_{\theta })\longrightarrow \fancyscript{U}_{\theta }, \quad \Xi \longmapsto \Xi _{\tau }\diamond \Xi \), is an isometry topological isomorphism.
Recall that, for \(p\in \mathbb {N}\) and \(\gamma _{1}\), \(\gamma _{2}\) \(>0\), we define the Hilbert space
where
Put \( G_\theta (N'\oplus N') = \bigcup _{p\in \mathbb {N}, \gamma _1,\gamma _2>0} G_{\theta , \gamma _1,\gamma _2}(N_{-p}\oplus N_{-p})\). The space \(G_{\theta }(N'\oplus N')\) carries the dual topology of \(F_\theta (N\oplus N)\) with respect to the \(\mathbb {C}\)-bilinear pairing given by \( \langle \!\langle \overrightarrow{\Phi }, \overrightarrow{\varphi }\,\rangle \!\rangle = \sum _{l,m=0}^{\infty } l!m! \langle \Phi _{l,m},\varphi _{l,m}\rangle \), where \(\overrightarrow{\Phi }=(\Phi _{l,m})_{l,m=0}^\infty \in G_{\theta }(N'\oplus N')\) and \(\overrightarrow{ \varphi } = (\varphi _{l,m})_{l,m=0}^\infty \in F_\theta (N\oplus N)\). One can easily see that \(\fancyscript{U}_{\theta }^{*}\) is given by
and we have the following isomorphism
Theorem 6
The map \(f_{\tau }\) defined by \(f_{\tau }: \fancyscript{L}(\fancyscript{F}_{\theta },\fancyscript{F}^{*}_{\theta })\longrightarrow \fancyscript{U}_{\theta }^{*}, \quad \Xi \longmapsto \Xi _{\tau }\diamond \Xi \), is an isometry topological isomorphism.
Theorem 7
The Wick symbol map realizes a topological isomorphism between the space \(\fancyscript{U}_{\theta }^{*}\) and the space \(\fancyscript{G}_{\theta ^{*}}(N\oplus N)\).
3 QWN-integral kernel operator
For \(x_{1}\), \(x_{2}\), \(z\) \(\in N\) and \(g(x_{1}, x_{2})\) in \(\fancyscript{H}_{\theta }(N\oplus N)\), let
Then, in view of Theorem 2, we give in the next theorem and proposition an analytic characterization of the QWN-derivatives and their adjoints, see also [6] and [24].
Theorem 8
Let be given \(z \in N\). For all \(\,\Xi \in \, \fancyscript{U}_{\theta }\), there exist a unique \(\widetilde{\Xi }_{1,z}\) and a unique \(\widetilde{\Xi }_{2,z}\) in \(\fancyscript{U}_{\theta }\) given by \(\widetilde{\Xi }_{1,z}=\omega ^{-1}\partial _{1,z}\omega (\Xi )\) and \(\widetilde{\Xi }_{2,z}=\omega ^{-1}\partial _{2,z}\omega (\Xi )\). Moreover, we have \(D_{z}^{-}\Xi =\widetilde{\Xi }_{1,z}\) and \(D_{z}^{+}\Xi =\widetilde{\Xi }_{2,z}\).
Proposition 1
For all \(\,\Xi \in \, \fancyscript{U}^{*}_{\theta }\) and all \(z \in N\), there exist a unique \(\widetilde{\Xi }^{*}_{1,z}\in \, \fancyscript{U}^{*}_{\theta }\) and a unique \(\widetilde{\Xi }^{*}_{2,z}\in \, \fancyscript{U}^{*}_{\theta }\) given by \(\widetilde{\Xi }^{*}_{1,z}=\omega ^{-1}\partial ^{*}_{1,z}\omega (\Xi )\) and \(\widetilde{\Xi }^{*}_{2,z}=\omega ^{-1}\partial ^{*}_{2,z}\omega (\Xi )\) where \(\partial ^{*}_{1,z}\) and \(\partial ^{*}_{2,z}\) are given by
for all \(g\in \fancyscript{H}_{\theta }(N\oplus N)\) and \(f\in \fancyscript{G}_{\theta ^{*}}(N\oplus N)\). Moreover, the operators \(\widetilde{\Xi }^{*}_{1,z}\) and \(\widetilde{\Xi }^{*}_{2,z}\) are denoted by \((D_{z}^{-})^{*}\Xi :=\widetilde{\Xi }^{*}_{1,z}\) and \((D_{z}^{+})^{*}\Xi :=\widetilde{\Xi }^{*}_{2,z}\).
From Theorems 2, 8 and Proposition 1, \(D_{z}^{\pm }\) is a continuous linear operator from \(\fancyscript{U}_{\theta }\) into itself, \((D_{z}^{\pm })^{*}\) is a continuous linear operator from \(\fancyscript{U}^{*}_{\theta }\) into itself and in particular, the restriction of \((D_{z}^{\pm })^{*}\) is a continuous linear operator from \(\fancyscript{U}_{\theta }\) into itself.
Definition 1
Let \(S=\sum _{l,m}\Xi _{l,m}(s_{l,m})\) in \(\fancyscript{U}_{\theta }\) and \(T=\sum _{l,m}\Xi _{l,m}(t_{l,m})\) in \(\fancyscript{U}^{*}_{\theta }\), where \(t_{l,m}\in (N')^{\widehat{\otimes } l+m}\) and \(s_{l,m}\in N^{\widehat{\otimes } l+m}\). Then, the duality between the two spaces \(\fancyscript{U}_{\theta }\) and \(\fancyscript{U}^{*}_{\theta }\), denoted by \(\langle \!\langle \!\langle .,. \rangle \!\rangle \!\rangle \), is defined as follows
Lemma 1
For \(S,T\in \fancyscript{U}_{\theta }\), we put
Then, there exist \(M(j,k,l,m)>0\) such that for any \(\alpha \ge 0\), \(\gamma _{1}\), \(\gamma _{2}\), \(\gamma _{3}\), \(\gamma _{4}>0\), we have
In particular, \(\eta _{_{S,T}}\in N^{\otimes (j+k+l+m)}\).
Proof
Let be given \(l,m\ge 0\) and \(\kappa _{l,m}\) in \(\left( (N^{\otimes l})\otimes (N^{\otimes m})\right) _{sym (l,m)}\). For \(z\in N\), by direct computation, the partial derivatives of the identity (8) in the direction \(z\) are given by
and
where for \(z_{p}\in (N^{\otimes p})'\), and \(\xi _{l+m-p}\in N^{\otimes (l+m-p)}\), \(p\le l+m\), the contractions \(z_{p}\otimes _{p}\kappa _{l,m}\) and \(\kappa _{l,m}\otimes ^{p}z_{p}\) are given by
Similarly, using Proposition 1 we get
Now, let \(S=\sum _{l,m}\Xi _{l,m}(s_{l,m})\) and \(T=\sum _{l,m}\Xi _{l,m}(t_{l,m})\) in \(\fancyscript{U}_{\theta }\) where \(s_{l,m},t_{l,m}\in N^{\widehat{\otimes } l+m}\). It then follows from (16) and (17) that
and
Hence,
Therefore,
Since, for \(\alpha \ge 0\) and \(\gamma _{1}, \gamma _{2}>0\)
we get
where \(\overrightarrow{f}=(s_{l,m})_{l,m}\), \(\overrightarrow{g}=(t_{j,k})_{j,k}\),
and
\(M_{1}<\infty \), see [23], which completes the proof. \(\square \)
Theorem 9
For any \(\kappa \in (N^{\otimes (j+k+l+m)})'\) there exists a continuous linear operator \(\Xi _{j,k,l,m}(\kappa )\in \fancyscript{L}(\fancyscript{U}_{\theta }, \fancyscript{U}^{*}_{\theta })\) such that
where
Moreover, for any \(\alpha \ge 0\), \(\gamma _{1}, \gamma _{2}>0\) with \(|\kappa |_{-\alpha }<\infty \) it holds that
Proof
First note that for any \(\kappa \in \big (N^{\otimes (j+k+l+m)}\big )'\),
is a continuous bilinear form on \(\fancyscript{U}_{\theta }\). In fact, by Lemma 1 we have
where \(M(j,k,l,m)\) is given in Lemma 1. Therefore, there is a continuous linear operator \(\Xi _{j,k,l,m}(\kappa )\) in \(\fancyscript{L}(\fancyscript{U}_{\theta },\fancyscript{U}^{*}_{\theta })\) such that
It then follows from (21) that
which completes the proof. \(\square \)
The operator \(\Xi _{j,k,l,m}(\kappa )\) is thus defined through two canonical bilinear forms:
where \(S,T \in \fancyscript{U}_{\theta }\). This suggests us to employ a formal integral expression:
We call \(\Xi _{j,k,l,m}\) an QWN-integral operator with kernel distribution \(\kappa \). It is possible to write down the action of \(\Xi _{j,k,l,m}(\kappa )\) explicitly using the contraction of tensor product.
For a later use we need to define an the operator \(S^{a,b}\) as follows. For any \(a,\, b\in N'\)
where \(\displaystyle {\kappa _{l,m}(a,b)=\frac{1}{l!m!}a^{\otimes l}\otimes b^{\otimes m}}\). It is noteworthy that \(\{S^{a,b};\; a,\,b \in N'\}\) spans a dense subspace of \(\fancyscript{U}^{*}_{\theta }\) and \(\{S^{a,b};\; a,\,b \in N\}\) spans a dense subspace of \(\fancyscript{U}_{\theta }\).
Proposition 2
Let \(S\in \fancyscript{U}_{\theta }\) given by \(S=\sum _{p,q=0}^{\infty }\Xi _{p,q}(s_{p,q})\). Then, for \(\kappa \in \big (N^{\otimes (j+k+l+m)}\big )'\) we have
In particular, for \(a,b\in N\) we have
Proof
Let \(T\in \fancyscript{U}_{\theta }\) be given as \(T=\sum _{p,q=0}^{\infty }\Xi _{p,q}(t_{p,q})\). Then, by definition,
from which (23) follows. The proof of (24) is then immediate. \(\square \)
During the above discussion we have obtained a linear map
But this not injective, namely, the kernel distribution is not uniquely determined.
For the uniqueness, we need a “partially symmetrized” kernel. We put
where \(.^{\pi }\) is defined by (see[21]), for \(F\in (N^{\otimes n})'\) and \(\pi \in G_{n}\)
We first note the following
Proposition 3
For all \(\kappa \in \big (N^{\otimes (j+k+l+m)}\big )'\) we have
Proof
Since \(\Xi _{j,k,l,m}(\kappa )\) is defined uniquely by (22), the assertion follows immediately from the fact that
which can be shown by a straight forward calculus. \(\square \)
Proposition 4
Let \(\kappa \in \big (N^{\otimes (j+k+l+m)}\big )'\). Then \(\Xi _{j,k,l,m}(\kappa )=0\) if and only if \(\widehat{\kappa }=0\). In other words, for \(\kappa \in \big (N^{\otimes (j+k+l+m)}\big )'_{sym(j,k,l,m)}\), the map
is injective.
Proof
Suppose that \(\widehat{\kappa }=0\). Then, we see from Proposition 3 that \(\Xi _{j,k,l,m}(\kappa )=\Xi _{j,k,l,m}(\widehat{\kappa })=0\). Conversely, we suppose that \(\Xi _{j,k,l,m}(\kappa )=0\). Consider a particular operator:
Then, it follows from the definition that
Since \(\Xi _{j,k,l,m}(\kappa )=0\) by assumption,
This being true for all, \(a,b,c,d\in N\), we conclude that \(\widehat{\kappa }=0\). \(\square \)
Proposition 5
Let \(\kappa \in \big (N^{\otimes (j+k+l+m)}\big )'\) and \(\kappa '\in \big (N^{\otimes (j'+k'+l'+m')}\big )'\). If \(\Xi _{j,k,l,m}(\kappa )=\Xi _{j',k',l',m'}(\kappa ')\ne 0\), then \(j=j'\), \(k=k'\), \(l=l'\), \(m=m'\) and \(\widehat{\kappa }=\widehat{\kappa }'\).
Proof
Suppose that \(\Xi _{j,k,l,m}(\kappa )=\Xi _{j',k',l',m'}(\kappa ')\ne 0\). In particular, since \(\Xi _{j,k,l,m}(\kappa )\ne 0\), it follows from Proposition 4 that \(\widehat{\kappa }\ne 0\) and therefore, there exist \(a,b,c,d\in N\) such that
Then, we have
On the other hand, unless \(j'\le j\), \(k'\le k\), \(l'\le l\) and \(m'\le m\),
Therefore, to have (25) it is necessary that \(j'\le j\), \(k'\le k\), \(l'\le l\) and \(m\le m'\). A similar argument with \(\Xi _{j',k',l',m'}(\kappa ')\ne 0\) implies that \(j\le j'\), \(k\le k'\), \(l\le l'\) and \(m\le m'\). Hence \(j'= j\), \(k'= k\), \(l'= l\) and \(m= m'\). We then see from Proposition 4 that \(\widehat{\kappa }=\widehat{\kappa '}\). \(\square \)
With each \((\kappa _{j,k,l,m})^{\infty }_{j,k,l,m=0}\in \bigoplus ^{\infty }_{j,k,l,m=0}\big (N^{\otimes (j+k+l+m)}\big )'_{sym(j,k,l,m)}\) (algebraic direct sum) we may associated an operator
It then follows from Propositions 4 and 5 that we have a linear injection:
4 QWN-symbol map
Definition 2
For \(P\in \fancyscript{L}(\fancyscript{U}_{\theta },\fancyscript{U}^{*}_{\theta })\), we define \(\sigma ^{Q}(P)\) to be the four-variable function given by
\(\sigma ^{Q}\) is referred to us a QWN-version of the usual symbol map \(\sigma \).
Note that \(\sigma ^{Q}(I)\) is given by
As in [10] and [17], we can define
Theorem 10
The map \(\sigma ^{Q}\) is a topological isomorphism from \(\fancyscript{L}(\fancyscript{U}_{\theta },\fancyscript{U}^{*}_{\theta })\) into \(\fancyscript{G}_{\theta ^{*}}(N^{4})\).
Proof
By the kernel theorem, we have
so that the QWN-symbol map can be seen as a composition of topological isomorphisms. \(\square \)
The QWN-symbol of a QWN-integral kernel QWN-operator is given in the following.
Proposition 6
Let \(\kappa \in \big (N^{\otimes (j+k+l+m)}\big )'\). Then, for \(a,b,c,d \in N\)
Proof
By Proposition 2 we have
On the other hand,
Hence
This completes the proof. \(\square \)
5 Chaotic expansion of QWN-operators:
Given \(\Xi ^{Q}\in \fancyscript{L}(\fancyscript{U}_{\theta },\fancyscript{U}^{*}_{\theta })\), we consider the Taylor expansion of \(\sigma ^{Q}(\Xi ^{Q})\) in \(\fancyscript{G}_{\theta ^{*}}(N^{4})\):
It is obvious by Theorem 10 that there exists \(\Xi _{j,k,l,m}\in \fancyscript{L}(\fancyscript{U}_{\theta },\fancyscript{U}^{*}_{\theta })\) such that
Thus, we come to
which is called the chaotic expansion of \(\Xi ^{Q}\in \fancyscript{L}(\fancyscript{U}_{\theta },\fancyscript{U}^{*}_{\theta })\).
Lemma 2
For each \(\kappa \in \big (N^{\otimes (j+k+l+m)}\big )'\) there exists a QWN-operator \(\Xi _{j,k,l,m}(\kappa )\) in \(\fancyscript{L}(\fancyscript{U}_{\theta },\fancyscript{U}^{*}_{\theta })\) whose symbol is given by
Proof
We write \(\Theta (a,b,c,d)\) for the righthand side of (26). It is sufficient to show that \(\Theta \in \fancyscript{G}_{\theta ^{*}}(N^{4})\) by Theorem 10. Since \(\big \langle \kappa , c^{\otimes j}\otimes d^{\otimes k}\otimes a^{\otimes l}\otimes b^{\otimes m}\big \rangle \) is of polynomial growth, it belongs to \(\fancyscript{G}_{\theta ^{*}}(N^{4})\). From the fact \(\fancyscript{U}_{\theta }\subset \fancyscript{U}^{*}_{\theta }\) we see that the identity operator I on \(\fancyscript{U}_{\theta }\) is a member of \(\fancyscript{L}(\fancyscript{U}_{\theta },\fancyscript{U}^{*}_{\theta })\), and hence \(\sigma ^{Q}(I)\in \fancyscript{G}_{\theta ^{*}}(N^{4})\). Note that
Since \(\fancyscript{G}_{\theta ^{*}}(N^{4})\) is closed under pointwise multiplication, we conclude that \(\Theta \in \fancyscript{G}_{\theta ^{*}}(N^{4})\). \(\square \)
Theorem 11
Let \(\Theta :N^{4}\rightarrow \mathbb {C}\) belongs to \(\fancyscript{G}_{\theta ^{*}}(N^{4})\). Then, there exists a unique family of kernel distributions \(\{\kappa _{j,k,l,m}\}^{\infty }_{j,k,l,m=0}\), where \(\kappa _{j,k,l,m}\in \big (N^{\otimes (j+k+l+m)}\big )'_{sym(j,k,l,m)}\), such that
for some \(K,\gamma _{1}, \gamma _{2}, \gamma _{3}, \gamma _{4}>0\) and
Moreover, the series
converges in \(\fancyscript{U}_{\theta }\), \(\Xi ^{Q}\in \fancyscript{L}(\fancyscript{U}_{\theta }, \fancyscript{U}^{*}_{\theta })\) and \(\sigma ^{Q}(\Xi ^{Q})=\Theta \).
For the proof we need some technical results.
Lemma 3
Let f be an entire holomorphic function on \(\mathbb {C}^{4}\) with the Taylor expansion \(f(z,y, x, w)=\sum _{j,k,l,m=0}^{\infty }a_{jklm}z^{j}y^{k}x^{l}w^{m}\). Assume that
for some \(K\ge 0\), \(K_{1}\ge 0\), \(K_{2}\ge 0\), \(K_{3}\ge 0\) and \(K_{4}\ge 0\). Then
Proof
Similar to the classical case (see[10, 21]). Since for \(R_{1}>0\), \(R_{2}>0\), \(R_{3}>0\) and \(R_{4}>0\)
we have
Hence, we obtain the desired estimate. \(\square \)
Lemma 4
Let \(\Theta \) be a \(\mathbb {C}-\)valued function on \(N^{4}\) and assume that \(\Theta \in \fancyscript{G}_{\theta ^{*}}(N^{4})\). Put
and
where
Then, \(\kappa _{j,k,l,m}\) is a continuous \((j+k+l+m)\)-linear form on N with \(\widehat{\kappa }_{j,k,l,m}=\kappa _{j,k,l,m}\). Moreover, there exist \(K,\gamma _{1}, \gamma _{2}, \gamma _{3}, \gamma _{4}>0\) such that
Proof
It is obvious that \(\kappa _{j,k,l,m}\) is a \(\mathbb {C}-\)valued \((j+k+l+m)-\)form on N. We now put
Then, we have the Taylor expansion:
For \(\rho _{1}\), \(\rho _{2}>0\) we have
Then from the facts that \(\theta \in \fancyscript{G}_{\theta ^{*}}(N^{4})\) and \(\theta ^{*}(s)+\theta ^{*}(t)\le \theta ^{*}(2s+2t)\), there exist \(K\ge 0\), \(\gamma _{1}\), \(\gamma _{2}\), \(\gamma _{3}\), \(\gamma _{4}>0\) such that
It then follows from Lemma 3 that
By virtue of the polarization formula (see[19, 21]) we obtain
Then, we have
where \(e\big (i^{(1)}\big )=e_{i_{1}^{(1)}}\otimes \cdots \otimes e_{i_{j}^{(1)}}\), \(e\big (i^{(2)}\big )=e_{i_{1}^{(2)}}\otimes \cdots \otimes e_{i_{k}^{(2)}}\), \(e\big (i^{(3)}\big )=e_{i_{1}^{(3)}}\otimes \cdots \otimes e_{i_{l}^{(3)}}\) and \(e\big (i^{(4)}\big )=e_{i_{1}^{(4)}}\otimes \cdots \otimes e_{i_{m}^{(4)}}\). Then we get
This completes the proof of (31). In particular, \(\kappa _{j,k,l,m}\in \big (N^{\otimes (j+k+l+m)}\big )'\). It is obvious that \(\widehat{\kappa }_{j,k,l,m}=\kappa _{j,k,l,m}\), namely \(\kappa _{j,k,l,m}\in \big (N^{\otimes (j+k+l+m)}\big )'_{sym(j,k,l,m)}\). \(\square \)
Proof
(of Theorem 11) The equality (27) is obvious from Lemma 4. We next prove identity (29). It follows from Proposition 6 that
Hence, in view of (32),
and therefore,
It follows from the uniqueness of the Taylor coefficients that \(\{A_{j,k,l,m}\}^{\infty }_{j,k,l,m=0}\) is unique, and therefore so is \(\{\kappa _{j,k,l,m}\}^{\infty }_{j,k,l,m=0}\) under the condition that \(\widehat{\kappa }_{j,k,l,m}=\kappa _{j,k,l,m}\). We then prove that \(\sum _{j,k,l,m=0}^{\infty }\Xi _{j,k,l,m}(\kappa _{j,k,l,m})S\) converges in \(\fancyscript{U}^{*}_{\theta }\) for any \(S\in \fancyscript{U}_{\theta }\). It follows from Theorem 9 that
In view of (31) we obtain
Since \(\theta _{n}\theta _{n}^{*}=(\frac{e}{n})^{n}\) for every \(n\ge 0\), see [23], then the series
converges for any \(S\in \fancyscript{U}_{\theta }\) if we chose \(\gamma _{1}\), \(\gamma _{2}>0\) such that
Then, \(\exists C\ge 0\) such that
This means that the series (29) converges in \(\fancyscript{U}^{*}_{\theta }\) and \(\Xi ^{Q}\in \fancyscript{L}(\fancyscript{U}_{\theta },\fancyscript{U}^{*}_{\theta })\). Finally we see that
For all \(a,b,c,d\in N\) as desired. \(\square \)
Theorem 12
For any \(\Xi ^{Q}\in \fancyscript{L}(\fancyscript{U}_{\theta },\fancyscript{U}^{*}_{\theta })\) there exists a unique family of distributions \(\kappa _{j,k,l,m} \in \big (N^{\otimes (j+k+l+m)}\big )'_{sym(j,k,l,m)}\), such that
where the righthand side converges in \(\fancyscript{U}^{*}_{\theta }\).
Proof
For a given \(\Xi ^{Q}\in \fancyscript{L}(\fancyscript{U}_{\theta },\fancyscript{U}^{*}_{\theta }) \) we put
Then, by Theorem 10 we see that \(\Theta \in \fancyscript{G}_{\theta ^{*}}(N^{4})\). Therefore, by Theorem 11 there exists a unique family of kernels \(\{\kappa _{j,k,l,m}\}^{\infty }_{j,k,l,m=0}\), \(\kappa _{j,k,l,m}\in (N^{\otimes (j+k+l+m)})'_{sym(j,k,l,m)}\) such that
Furthermore, as is stated in Theorem 11,
converges in \(\fancyscript{U}^{*}_{\theta }\), \(\Xi '^{Q}\in \fancyscript{L}(\fancyscript{U}_{\theta },\fancyscript{U}^{*}_{\theta })\) and \(\sigma ^{Q}(\Xi '^{Q})(a,b,c,d)=\Theta (a,b,c,d)\) for all \(a,b,c,d\in N\). The last identity and (33) yield
Since \(\{S^{a,b}, \;\ a,b \in N\}\) span a dense subspace of \(\fancyscript{U}_{\theta }\) and both \(\Xi ^{Q}\) and \(\Xi '^{Q}\) are continuous, we conclude that \(\Xi ^{Q}=\Xi '^{Q}\). \(\square \)
Example 1
For \(z\in N'\), the QWN-derivatives and their adjoints studied in [16] and [6] coincide respectively with
Example 2
For \(K_{1}, K_{2}\in \fancyscript{L}(N',N')\), the QWN-Laplacians studied in [6] (see also [15]) and their adjoints are given by
Example 3
For \(K_{1}, K_{2},B_{1},B_{2}\in \fancyscript{L}(N',N')\), the QWN-Fourier Gauss transform studied in [15] and [6] is given by
where \(\kappa \) is given by
Example 4
For \(c,d\in N\), the QWN-translation operator admits the following representation (see [5])
Example 5
Let \(S=\sum _{i,j=0}^{\infty }\Xi _{i,j}(s_{i,j})\in \fancyscript{U}^{*}_{\theta }\). Then, the QWN-convolution operator \(C_{S}^{Q}\), defined in [5], coincides with
6 QWN-rotation group
Let \(O(X, H)\) given by (see [21])
which is called infinite-dimensional rotation group. We say that a continuous operator from \(\fancyscript{U}_{\theta }\) into \(\fancyscript{U}^{*}_{\theta }\) is rotation-invariant if
where \(\Gamma ^{Q}(B)\) is given by
for \(\Xi =\sum _{l,m}\Xi _{l,m}(\kappa _{l,m})\) in \(\fancyscript{U}_{\theta }\). By definition, if \(\Xi ^{Q}\) is rotation invariant, so is \((\Xi ^{Q})^{*}\).
The main purpose of this section is to characterize all rotation-invariant operators \(\Xi ^{Q}\) in terms of the QWN-Gross Laplacian and the QWN-conservation operator given by
We recall that (see [21]) \(F\in (N^{\otimes n})'\) is rotation-invariant if \((B^{\otimes n})^{*}F=F\) for all \(B\in O(X,H)\).
Theorem 13
Let \(\Xi ^{Q}\in \fancyscript{L}(\fancyscript{U}_{\theta }, \fancyscript{U}^{*}_{\theta })\) and \(\Xi ^{Q}=\sum ^{\infty }_{j,k,l,m=0}\Xi _{j,k,l,m}(\kappa )\), where \(\kappa \in \big (N^{\otimes (j+k+l+m)}\big )'_{sym(j,k,l,m)}\). Then \(\Xi ^{Q}\) is rotation invariant if and only if \(\kappa \) is rotation invariant.
Proof
Let \(\Xi ^{Q}\in \fancyscript{L}(\fancyscript{U}_{\theta }, \fancyscript{U}^{*}_{\theta })\) given by \(\Xi ^{Q}=\sum _{j,k,l,m=0}^{\infty }\Xi _{j,k,l,m}(\kappa )\), where \(\kappa \in \big (N^{\otimes (j+b+l+m)}\big )'_{sym(j,k,l,m)}\). Recall that \(\Gamma ^{Q}(B)\Xi ^{a,b}=\Xi ^{Ba,Bb}, \;\ a,b\in N\). Then, for any \(a,b\in N\) we have
From Proposition 6, we obtain
Then, using the obvious equality, for \(\Xi ^{Q}=\sum _{j,k,l,m}\Xi _{j,k,l,m}(\kappa )\)
We get
In particular, \((\Gamma ^{Q}(B))^{*}\Xi _{j,k,l,m}(x)P^{Q}(B)=\Xi _{j,k,l,m}\big (\big (B^{\otimes (j+k+l+m)}\big )^{*}\kappa \big )\). Therefore, by the uniqueness of the Fock expansion, \(\Xi ^{Q}\) is rotation invariant if and only if \(\Xi _{j,k,l,m}(\kappa )\) is rotation invariant if and only if \(\kappa \) is rotation invariant. \(\square \)
Let \(\Delta _{G}^{Q_{+}}\) and \(\Delta _{G}^{Q_{-}}\) given by
Lemma 5
Let \(\alpha ', \alpha ^{''}\), \(\beta ', \beta ^{''}\), \(\lambda ', \lambda ^{''}\) be non-negative integers and put \(j=2\alpha '+\beta '\), \(k=2\alpha ^{''}+\beta ^{''}\), \(l=2\lambda '+\beta '\) and \(m=2\lambda ^{''}+\beta ^{''}\). Then,
where \(\tau _{\beta '}\in (X^{\otimes 2\beta '})'\) is given by
Proof
Since both sides of (36) are continuous operators from \(\fancyscript{U}_{\theta }\) into \(\fancyscript{U}^{*}_{\theta }\), it suffices to check that they coincide on the operator \(\Xi ^{a,b}\), for \(a,b\in N\), or by applying the QWN-symbol map. To this end, let a,b,c,d \(\in N\), then since
we get,
On the other hand, from the commutation between \((D^{-}_{s})^{*}\) and \(D_{t}^{+}\) we have
Hence we complete the proof. \(\square \)
Theorem 14
Let \(\kappa \in \big (N^{\otimes (j+k+l+m)}\big )'\) and assume that \(\Xi _{j,k,l,m}(\kappa )=0\) is rotation invariant. If \(j+k+l+m\) is odd, then \(\Xi _{j,k,l,m}(\kappa )=0\). If \(j+k+l+m\) is even, then \(\Xi _{j,k,l,m}(\kappa )\) is a linear combination of \(\big (\big (\Delta _{G}^{Q}\big )^{\alpha }\big )^{*}(N^{Q})^{\beta }\big (\Delta _{G}^{Q}\big )^{\lambda }\) with \(\alpha \), \(\beta \), \(\lambda \) being non-negative integers such that \(\alpha +\beta +\lambda \le (j+k+l+m)/2\).
Proof
Suppose that \(\Xi _{j,k,l,m}(\kappa )\) is rotation invariant. Without loss of generality we may assume that \(\kappa \in (N^{\otimes (j+k+l+m)})'_{sym(j,k,l,m)}\). Then, \(\kappa \) is rotation invariant by Theorem 13. It is well known (see [21]) that if \(F\in (N^{\otimes n})'\) is rotation invariant and n is odd then \(F=0\) and if \(n=2p\) then F is a linear combination of \(\big (\tau ^{\otimes p}\big )^{\sigma }\) for \(\sigma \in G_{n}\). From which we deduce that if \(j+k+l+m\) is odd then \(\kappa =0\) and hence \(\Xi _{j,k,l,m}(\kappa )=0\).
We consider now the case when \(j+k+l+m\) is even. Then \(\kappa \) is a linear combination of \((\tau ^{\otimes (j+k+l+m)/2})^{\sigma }\), \(\sigma \in G_{j+k+l+m}\). For each \(\sigma \in G_{j+k+l+m}\) we may find \(\sigma '\in G_{j}\times G_{k}\times G_{l}\times G_{m}\) such that
for some non-negative integers \(\alpha '\), \(\alpha ''\), \(\beta '\), \(\beta ''\), \(\lambda '\), \(\lambda ''\) with \(j=2\alpha '+\beta '\), \(k=2\alpha ^{''}+\beta ^{''}\), \(l=2\lambda '+\beta '\) and \(m=2\lambda ^{''}+\beta ^{''}\). Then in view of Lemma 5, we have
On the other hand
and similarly,
Then it suffices to prove that \(\Xi _{\beta ',\beta '', \beta ',\beta ''}(\tau _{\beta '}\otimes \tau _{\beta ''})\) is a polynomial in the QWN-conservation operator \(N^{Q}\) or in \(N^{Q_{+}}\) and \(N^{Q_{-}}\), where \(N^{Q}=N^{Q_{-}}+N^{Q_{+}}\), in other words it suffices to prove that
To this end, denote by P the righthand side of (37), then
Hence we complete the proof. \(\square \)
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Barhoumi, A., Ammou, B.K.B. & Rguigui, H. Operator theory: quantum white noise approach. Quantum Stud.: Math. Found. 2, 221–241 (2015). https://doi.org/10.1007/s40509-015-0033-y
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DOI: https://doi.org/10.1007/s40509-015-0033-y
Keywords
- QWN-derivatives
- QWN-operators
- QWN-symbol map
- Rotation-invariant QWN-operators
- QWN-Gross Laplacian
- QWN-conservation operator