Abstract
We investigate the group \({\mathcal {H}}_{\mathbb {C}}\) of complexified Heisenberg matrices with entries from an infinite-dimensional complex Hilbert space H. Irreducible representations of the Weyl–Schrödinger type on the space \(L^2_\chi \) of quadratically integrable \({\mathbb {C}}\)-valued functions are described. Integrability is understood with respect to the projective limit \(\chi =\varprojlim \chi _i\) of probability Haar measures \(\chi _i\) defined on groups of unitary \(i\times i\)-matrices U(i). The measure \(\chi \) is invariant under the infinite-dimensional group \(U(\infty )=\bigcup U(i)\) and satisfies the abstract Kolmogorov consistency conditions. The space \(L^2_\chi \) is generated by Schur polynomials on Paley–Wiener maps. The Fourier-image of \(L^2_\chi \) coincides with the Hardy space \({H}^2_\beta \) of Hilbert–Schmidt analytic functions on H generated by the correspondingly weighted Fock space \(\varGamma _\beta (H)\). An application to heat equation over \({\mathcal {H}}_{\mathbb {C}}\) is considered.
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1 Introduction
An aim of this work is to investigate irreducible Weyl–Schrödinger representations of the complexified Heisenberg group \({\mathcal {H}}_\mathbb {C}\) (see [17, n.9]), consisting of matrix elements X(a, b, t) with any \(a,b\in {H}\) and \(t\in {\mathbb {C}}\) such that
where H is an infinite-dimensional complex Hilbert space and \(\mathbb {1}\) is its identity map.
The group \({\mathcal {H}}_\mathbb {C}\) has the unit X(0, 0, 0) and inverse elements of the form \(X(a,b,t)^{-1}=X\left( -a,-b,-t+\langle {a}\mid {b}\rangle \right) \).
In what follows, we consider the infinite-dimensional unitary group \(U(\infty )=\bigcup {U}(i)\), containing all subgroups U(i) of unitary \(i\times i\)-matrices, which acts irreducibly on a complex Hilbert space \(\left\{ H,\langle \cdot \mid \cdot \rangle \right\} \) with an orthonornal basis \(({\mathfrak {e}}_i)_{i\in {\mathbb {N}}}\).
To find the desired representation, we use the space \(L^2_\chi \) of \({\mathbb {C}}\)-valued functions that are quadratically integrable with respect to the probability measure \(\chi \). Wherein, according to our assumption \(\chi \) has a structure of the projective limit \(\chi =\varprojlim \chi _i\) of probability Haar’s measures \(\chi _i\) on U(i), satisfying the Kolmogorov consistency conditions in an abstract Bochner’s formulation (see [23, 27]).
In [21, 24] it was shown that the projective limit \(\chi =\varprojlim \chi _i\) is well defined over the projective limit \({\mathfrak {U}}=\varprojlim U(i)\) with respect to the Livšic transforms \(\pi _i^{i+1}:{U(i+1)\rightarrow U(i)}\) such that \(\chi _i=\pi _i^{i+1}(\chi _{i+1})\). In this paper, we prove that for such \(\chi \) each function from \(L^2_\chi \) admit a superposition (linearization in the sense of [5]) on Paley–Wiener maps associated with \(U(\infty )\). As a result, it is shown that Schur polynomials form an orthonormal basis in \(L^2_\chi \) and the Fourier-image of \(L^2_\chi \) consists of Hilbert-Schmidt analytic functions on H.
Note also that projective limits of probability measures over various infinite-dimensional manifolds with similar properties were investigated in [25, 34, 35].
If instead of the unitary group \(U (\infty )\) we take the infinite-dimensional linear space with a Gaussian measure \(\gamma \), a similar construction of the appropriate space \(L^2_\gamma \) can be found in the well-known works [1, 2]. In this case, the Fourier-image of \(L^2_\gamma \) coincides with the Segal–Bargmann space of entire analytic functions over which the Schrödinger type irreducible representations of Heisenberg groups are well defined. In the present paper, we change \(\gamma \) by the unitarily-invariant projective limit \(\chi =\varprojlim \chi _i\) and, as a result, we obtain another irreducible representation, called to be the Weyl–Schrödinger type.
Infinite-dimensional Heisenberg groups over \({\mathbb {R}}\) was considered in [19] by using the reproducing kernel Hilbert spaces. The Schrödinger representation of such groups using Gaussian measures over a real Hilbert space was described in [3]. Since the group \({\mathcal {H}}_\mathbb {C}\) in the case of matrix entries \(a,b,t\in {\mathbb {R}}\) coincides with the classical Heisenberg group over \({\mathbb {R}}\) (see, e.g. [11]), the results of the present paper can be considered as a complexification of previous studies. The Weyl–Schrödinger representation obtained here is not equivalent to that was described earlier.
Further, let us briefly describe the main results. Consider the following mapping \(\phi :H\ni {h}\longmapsto \phi _h\in L^2_\chi \) defined by Paley–Wiener maps
where \({\mathfrak {e}}^*_i(\cdot ):=\left\langle \cdot \mid {\mathfrak {e}}_i\right\rangle \) and the projections \(\pi _i:{\mathfrak {U}}\ni {\mathfrak {u}} \rightarrow u_i\in U(i)\) are uniquely defined by \(\pi _i^{i+1}\). Every function \(\phi _h\) of variable \({{\mathfrak {u}}\in {\mathfrak {U}}}\) satisfies the equality (Corollary 3)
The space \(L^2_\chi \) can be generated by two orthonormal bases, consisting of Schur polynomials and power polynomials of variables \({\phi }_\imath = \left( {\phi }_{\imath _1},\ldots ,{\phi }_{\imath _\eta }\right) \), respectively,
These bases are indexed by tabloids \(\imath ^\lambda \) with strictly ordered \(\imath =({\imath _1},\ldots ,{\imath _\eta })\in {\mathbb {N}}^\eta \) where \(\lambda =(\lambda _1,\ldots ,\lambda _\eta )\in {\mathbb {N}}^\eta \) is a partition of \(n\in {\mathbb {N}}\) and \(\eta =\eta (\lambda )\) stands for the length of \(\lambda \). Then we write briefly \({\imath ^\lambda \vdash n}\). The orthogonal expansion \(L_\chi ^2 =\bigoplus {L}_\chi ^{2,n}\) holds (Theorem 1) where \({L}_\chi ^{2,n}\) are formed by n-homogeneous polynomials \({\phi }^\lambda _\imath \), normed as follows
It is also shown that the surjective linear isometry \(\varPsi :{H}^2_\beta \ni \psi ^*_f\longmapsto {f}\in {L}^2_\chi \) holds (Lemma 5), where \({H}^2_\beta =\sum {P}_\beta ^n(H)\) means the Hardy space of entire analytic functions \(\psi ^*_f(h)\) of variable \(h\in H\) and \({P}_\beta ^n(H)\) is generated by the n-homogeneous Hilbert–Schmidt polynomials \({\mathfrak {e}}^{*\lambda }_\imath := {\mathfrak {e}}^{*\lambda _1}_{\imath _1}\ldots {\mathfrak {e}}^{*\lambda _\eta }_{\imath _\eta }\), normed as \(\Vert {\mathfrak {e}}^{*\lambda }_\imath \Vert _{{H}^2_\beta }= \big (\beta _\lambda {\lambda !}\big )^{1/2}\).
If the basis of symmetric tensor elements \({\mathfrak {e}}^{\odot \lambda }_\imath :={\mathfrak {e}}^{\otimes \lambda _1}_{\imath _1}\odot \ldots \odot {\mathfrak {e}}^{\otimes \lambda _\eta }_{\imath _\eta }\) (associated with \({\mathfrak {e}}^{*\lambda }_\imath \)) in the correspondingly weighted Fock space \(\varGamma _\beta (H)\) is normed as \(\Vert {\mathfrak {e}}^{\odot \lambda }_\imath \Vert _{\varGamma _\beta }= \Vert {\mathfrak {e}}^{*\lambda }_\imath \Vert _{{H}^2_\beta }\) then each function \(f\in {L}^2_\chi \) admits the superposition
where the Taylor expansion on the right-hand side of any analytic function \(\psi ^*_f\in {H}^2_\beta \) on H is uniquely determined by the corresponding element \(\psi _f\in \varGamma _\beta (H)\).
Our further goal is to analyze the inverse isomorphism \(\varPsi ^{-1}\) which can be described by the Fourier transform under the measure \(\chi \) in following way
The Fourier transform F acts isometrically on the Hardy space of analytic functions \({H}^2_\beta \) (Theorem 2). So, F acts as an analytic extension of the mapping \(\phi \).
Applying the superposition with \(\varPsi \), we describe two different representations of the additive group \((H,+)\) over \(L^2_\chi \) defined by shift and multiplicative groups (Lemma 7). Using this we show (in Theorem 3) that an irreducible representation of the Heisenberg group \({\mathcal {H}}_{\mathbb {C}}\) can be realized on \({L}^2_\chi \) in the Weyl–Schrödinger form
for all \(a,b\in H\) and \(z\in {\mathbb {C}}\), where \(T_b^\dagger \) and \(M_{a^*}^\dagger \) are defined by shift and multiplicative groups, respectively. It is also proved that the Weyl system \({W}^\dagger (a,b)\) has the densely-defined generator \({\mathfrak {p}}^\dagger _{a,b}:=\partial _{b}^\dagger +{{\bar{\phi }}}_{a}\) which satisfies the commutation relation
where the groups \(M_{a^*}^\dagger \) and \(T_b^\dagger \) are generated by \({{\bar{\phi }}}_a\) and \(\partial _b^\dagger \), respectively.
Applying the Weyl–Schrödinger representation to the associated with \({\mathcal {H}}_{\mathbb {C}}\) heat equation, we prove (Theorem 4) that the following Cauchy problem with \(\partial _i^\dagger :=\partial _{{\mathfrak {e}}_i}^\dagger \),
has the unique solution \(w(r)={\mathfrak {G}}^\dagger _rf\) for any function f from a finite sum \(\bigoplus {L}^{2,n}_\chi \), where the 1-parameter Gaussian semigroup \({\mathfrak {G}}_r^\dagger \) has the form
Here \(\tau =(\tau _i)\) belongs to the abstract Wiener space \(\{w_0,\Vert \cdot \Vert _{w_0}\}\) defined by the injections \(l_2\looparrowright {w}_0\looparrowright {c}_0\) of real Banach spaces and endowed with the Wiener measure \({\mathfrak {w}}\) in according to the known Gross’ theorem [10], whereas the sequence of projectors \((p^\sim _n)\) onto \({\mathbb {R}}^n\) is convergent to the identity map on \(w_0\).
Finally, note that this work is a continuation of previous publications [16, 17]. The novelty results from the observation that the system of Schur polynomials with variables on Paley–Wiener maps form an orthonormal basis in \(L^2_\chi \). This allowed us to investigate irreducible Weyl–Schrödinger representations and Weyl systems of the Heisenberg group \({\mathcal {H}}_{\mathbb {C}}\) on the whole space \(L^2_\chi \).
2 Invariant Probability Measure
Consider the unitary group \(U(\infty )=\bigcup U(m)\) with \(m\in {\mathbb {N}}_0={\mathbb {N}}\cup \{0\}\), \( \mathbb {1}=U(0)\), irreducibly acting on a separable Hilbert space H, where subgroups U(m) are identified with ranges of injections \(U(m)\ni u_m\longmapsto {\begin{bmatrix} u_m &{}\quad 0\\ 0 &{}\quad \mathbb {1}\\ \end{bmatrix}\in U(\infty )}\). Following to [21, 24], we use the Livšic transforms \(\pi ^{m+1}_m:U(m+1)\rightarrow U(m)\) of the form
with \(z_m\in U(m)\) defined by excluding \(x_1=y_1\in {\mathbb {C}}\) from \(\begin{bmatrix} y_m \\ y_1\\ \end{bmatrix}=\begin{bmatrix} z_m &{}\quad a\\ -b &{}\quad -t\\ \end{bmatrix} \begin{bmatrix} x_m \\ x_1\\ \end{bmatrix}\) for \(x_m,y_m\in {\mathbb {C}}^m\) and \(a,b\in {\mathbb {C}}\) [24, Lem. 3.1]. It is surjective (not continuous) Borel mapping [24, Lem. 3.11].
The projective limit \({\mathfrak {U}}:=\varprojlim U(m)\) under \(\pi ^{m+1}_m\) has surjective Borel (not group homomorphisms) projections
Their elements \({{\mathfrak {u}}\in {\mathfrak {U}}}\) are called the virtual unitary matrices. The right action
is defined to be \(\pi _m({\mathfrak {u}}.g)=w^{-1}\pi _m({\mathfrak {u}})v\), where m is large enough that \(v,w\in {U}(m)\). On \({\mathfrak {U}}\) the involution \({\mathfrak {u}}\mapsto {\mathfrak {u}}^\star =(u_k^\star )\) is well defined, where \(u_k^\star =u_k^{-1}\) is adjoint to \({u_k\in U(k)}\). Thus, \([\pi _m({\mathfrak {u}}.g)]^\star =\pi _m({\mathfrak {u}}^\star .g^\star )\) for all \(g^\star ={(w^\star ,v^\star )}\in {U(\infty )\times U(\infty )}\).
There exists the dense embedding \(U(\infty )\looparrowright {\mathfrak {U}}\) (see [24, n.4]) which assigns the stabilized sequence \({\mathfrak {u}}=(u_k)\) to each \({u_m\in U(m)}\) such that
We always assume that the group U(m) is endowed with the probability Haar measure \(\chi _m\). Using the Kolmogorov consistency theorem (see, e.g. [24, Lem.4.8], [27, Thm 2.2], [30, Cor.4.2]), we determine the probability measure on \({\mathfrak {U}}\) to be the projective limit
where \(\pi _m^{m+1}(\chi _{m+1})\) means an image-measure and \({\chi _0=1}\). As is known [30, Thm 2.5], the measure \(\chi \) is Radon. We now describe the necessary properties of \(\chi \).
Consider the Hilbert space \(L^2_\chi \) of functions \(f:{\mathfrak {U}}\rightarrow {\mathbb {C}}\) with the following norm and inner product
Let \(L_\chi ^\infty \) be the space of \(\chi \)-essentially bounded functions \(f:{\mathfrak {U}}\rightarrow {\mathbb {C}}\) with the norm \(\Vert f\Vert _\infty ={\mathop {\hbox {ess sup}}}_{{\mathfrak {u}}\in {\mathfrak {U}}}|f({\mathfrak {u}})|\). The embedding \(L^\infty _\chi \looparrowright L^2_\chi \) holds and \(\Vert f\Vert _\chi \le \Vert f\Vert _\infty \).
Lemma 1
For any \(f\in {L}_\chi ^\infty \) there exists the limit
Moreover, the measure \(\chi \) is invariant under the right action, which means that
Proof
The sequence \(\{(\chi _m\circ \pi _m)({\mathcal {K}})\}\) is decreasing for any compact set \({\mathcal {K}}\) in \({\mathfrak {U}}\), since \(\pi _m={\pi ^{m+1}_m\circ \pi _{m+1}}\) yields \(\pi _{m+1}({\mathcal {K}})\subseteq (\pi ^{m+1}_m)^{-1}\left[ \pi _m({\mathcal {K}})\right] \). It follows
This ensures that the necessary and sufficient conditions of the Prokhorov theorem [4, Thm IX.52] and its modification from [30, Thm 4.2] are satisfied.
Indeed, let \({\check{U}}(m)\subset U(m)\) be the set of matrices with no eigenvalue \(\{-1\}\) for \({m\ge 1}\). As is known [24, n.3], \({\check{U}}(m)\) is open in U(m) and \({\chi _m(U(m){\setminus }{\check{U}}(m))}=0\). In virtue of [24, Lem. 3.11] the restrictions \(\pi ^{m+1}_m:{\check{U}}(m+1)\rightarrow {\check{U}}(m)\) are continuous and surjective. The projective limit \(\varprojlim {\check{U}}(m)\) under these restrictions has continuous surjective projections \(\pi _m:\varprojlim {\check{U}}(m)\rightarrow {\check{U}}(m)\). Restrict \(\chi _m\) to \({\check{U}}(m)\). By [30, Thm 6], a probability measure \({\check{\chi }}\) satisfying conditions \(\pi _m({\check{\chi }})=\chi _m|_{{\check{U}}(m)}\) is well defined iff for every \(\varepsilon >0\) there exists a compact set \({\mathcal {K}}\subset \varprojlim {\check{U}}(m)\) such that
Then by the Prokhorov theorem \({\check{\chi }}\) is uniquely determined as
Let \(\varepsilon >0\) and \(K_1\subset {\check{U}}(1)\) be a compact set such that \(\chi _1(K_1)>1-\varepsilon \). Let a compact sets \(K_m\subset {\check{U}}(m)\) be defined inductively such that
Assume that \(K_1,\ldots ,K_m\) are constructed. Since \(\chi _m=\pi ^{m+1}_m(\chi _{m+1})\), we get
By regularity of \(\chi _{m+1}|_{{\check{U}}(m)}\), there exists a compact set
The induction is complete. Then \({\mathcal {K}}=\varprojlim K_m\) with \(K_0=\mathbb {1}\) is compact. By virtue of (10), we have \({{\check{\chi }}}({\mathcal {K}})\ge 1-\varepsilon \). Hence, the projective limit \({{\check{\chi }}}=\varprojlim \chi _m|_{{\check{U}}(m)}\) is well defined on \(\varprojlim {\check{U}}(j)\) by the Prokhorov criterion.
The measure \({\check{\chi }}\) can be extended to \(\varprojlim {U}(m){\setminus }\varprojlim {\check{U}}(m)\) as zero, since each \(\chi _m\) is zero on \({U(m){\setminus }{\check{U}}(m)}\). The uniqueness of the projective limits yields \({\check{\chi }}=\chi \). So, \(\chi =\varprojlim \chi _m\) is also well defined and by (9) and (10) we get
By the known Portmanteau theorem [14, Thm 13.16] it follows that the limit (6) exists. Whereas, the property (7) is a consequence of the equalities
for all \(g=(v,w)\in {U(\infty )\times U(\infty )}\) where m is large enough that \(v,w\in U(m)\).
Finally, the function \(({\mathfrak {u}},g)\mapsto f({\mathfrak {u}}.g)\) with any \({f\in {L}_\chi ^\infty }\) is integrable over \({{\mathfrak {U}}\times U(m)\times U(m)}\), hence
by the Fubini theorem. It yields (8) since the internal integral on the right-hand side is independent of g by (7) and \({\int \,d(\chi _m\otimes \chi _m)(g)=1}.\) The proof is complete. \(\square \)
We now note the concentration property of Haar measures sequence \((\chi _m)\) satisfying the Kolmogorov conditions \(\chi _m=\pi _m^{m+1}(\chi _{m+1})\) if each group U(m) is endowed with the normalized Hilbert–Schmidt metric
As is well known (see [9, 31]), \((U(m),d_{HB},\chi _m)\) is a Lévy family. Namely, the following sequence of isoperimetric constants dependent on \(\varepsilon >0\)
with \((\varOmega _m)_\varepsilon =\left\{ u_m\in U(m):d_{HS}\left( u_m,\varOmega _m\right) <\varepsilon \right\} \) is such that
Taking into account the Lemma 1, we can formulate the following conclusion.
Corollary 1
For any Borel set \(\varOmega _\varepsilon =\varprojlim (\varOmega _m)_\varepsilon \) with \(\chi _m(\varOmega _m)>1/2\) in the projective limit \({\mathfrak {U}}=\varprojlim U(m)\) the equality
holds. Consequently, all Borel sets \({\mathfrak {U}}{\setminus }\varOmega _\varepsilon \) with \({\chi _m(\varOmega _m)>1/2}\) and any \({\varepsilon >0}\) are \(\chi \)-measure zero, i.e., the measure \(\chi =\varprojlim \chi _m\) is concentrated outside these sets.
3 Polynomials on Paley–Wiener Maps
Let \({\mathscr {I}}_\eta :=\big \{\imath =\big ({\imath _1},\ldots ,{\imath _\eta }\big )\in {\mathbb {N}}^\eta :\imath _1<\imath _2<\ldots <\imath _\eta \big \}\) be an integer alphabet of length \(\eta \) and \({\mathscr {I}}=\bigcup {\mathscr {I}}_\eta \). Let \(\lambda =(\lambda _1,\ldots ,\lambda _\eta )\in {\mathbb {N}}^\eta \) with \({\lambda _1\ge \lambda _2\ge \ldots \ge \lambda _\eta }\) be a partition of an n-letter word \(\imath ^\lambda =\big \{\Box _{ij}:{1\le i\le \eta }, j=1,\ldots ,\lambda _i\big \}\) with \({\imath \in {\mathscr {I}}_\eta }\). A Young \(\lambda \)-tableau with a partition \(\lambda \) is a result of filling the word \(\imath ^\lambda \) onto the matrix \([\imath ^\lambda ]=\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} \Box _{11} &{} \dots &{} \dots &{}\Box _{1\lambda _1}\\ \vdots &{} \vdots &{} \ddots &{} \\ \Box _{\eta 1} &{} \dots &{}\Box _{\eta \lambda _\eta } &{} \end{array}\!\!\!\!\!\) with n nonzero entries in some way without repetitions. So, each \(\lambda \)-tableau \([\imath ^\lambda ]\) can be identified with a bijection \([\imath ^\lambda ]\rightarrow \imath ^\lambda \). The conjugate partition \(\lambda ^\intercal \) corresponds to the transpose matrix \([\imath ^\lambda ]^\intercal \).
A Young tableau \([\imath ^\lambda ]\) is called standard (semistandard ) if its entries are strictly (weakly) ordered along each row and strictly ordered down each column. Let \({\mathbb {Y}}\) denote all Young tabloids \([\imath ^\lambda ]\) and \({\mathbb {Y}}_n\) be its subset such that \(\imath ^\lambda \vdash n\). Assume that \({\mathbb {Y}}_0=\left\{ \emptyset \in {\mathbb {Y}}:|\emptyset |=0\right\} \) and \(\eta (\emptyset )=0\).
As before, \(\left\{ H,\langle \cdot \mid \cdot \rangle \right\} \) is a separable complex Hilbert space with an orthonormal basis \(\left\{ {\mathfrak {e}}_i:i\in {\mathbb {N}}\right\} \) and \({\Vert \cdot \Vert ={\langle \cdot \mid \cdot \rangle ^{1/2}}}\). For its adjoint space \(H^*\) the conjugate-linear isometry \({*:H^*\rightarrow H^{**}=H}\) is defined via \(a^*(h)={\langle h\mid a\rangle }\) for all \({a,h\in H}\). The Fourier expansion \(h=\sum {\mathfrak {e}}^*_i(h){\mathfrak {e}}_i\) with \({\mathfrak {e}}^*_i(h):={\langle h\mid {\mathfrak {e}}_i\rangle }\) holds. The tensor power \({H}^{\otimes n}\!,\) spanned by elements \(\psi _n={h_1\otimes \ldots \otimes h_n}\) with \({h_i\in {H}}\)\({(i=1,\ldots ,n)}\), is endowed with the norm \(\Vert \psi _n\Vert ={\left\langle \psi _n\mid \psi _n\right\rangle }^{1/2}\) where \({\left\langle \psi _n\mid \psi _n'\right\rangle }:= {\langle h_1\mid h'_1\rangle \ldots \langle h_n\mid h'_n\rangle }\).
Let \(S_n\) be the group of n-elements permutations \(\sigma (\psi _n):=h_{\sigma (1)}\otimes \ldots \otimes {h}_{\sigma (n)}\). An orthogonal basis in \({H}^{\otimes n}\) is formed by elements \(\sigma ({\mathfrak {e}}^{\otimes \lambda _1}_{\imath _1}\otimes \ldots \otimes {\mathfrak {e}}^{\otimes \lambda _\eta }_{\imath _\eta })\) with \({\imath ^\lambda \vdash n}\) and \(\eta =\eta (\lambda )\), additionally indexed by all \(\sigma \in S_n\). The symmetric tensor power \(H^{\odot n}\subset H^{\otimes n}\) is defined to be a range of the orthogonal projector \({{\mathcal {S}}_n:{H}^{\otimes n}\ni \psi _n\longmapsto {h_1\odot \ldots \odot h_n}:=(n!)^{-1}{\sum }_{\sigma \in S_n}\sigma (\psi _n)}\). We assume that \({H}^{\otimes n}\) is completed and that \({H}^{\otimes 0}={\mathbb {C}}\). Let \(\psi _n:=h^{\otimes n}\) for \({h=h_i}\). The embedding \(\left\{ h^{\otimes n}:{h}\in {H}\right\} \subset {H}^{\odot n}\) is total by the polarization formula [7, n.1.5]
Let \(H_\eta \subset H\) be spanned by \(\big \{{\mathfrak {e}}_{\imath _1},\ldots ,{\mathfrak {e}}_{\imath _\eta }\big \}\). We can uniquely assign to any semistandard tableau \([\imath ^\lambda ]\) with \(\imath ^\lambda \vdash n\) the element in \(H_\eta ^{\otimes n}\) for which there exists the permutation \({\sigma '\in {S}_n}\) such that \({\sigma '\big ({\mathfrak {e}}^{\otimes \lambda _1}_{\imath _1}\otimes \ldots \otimes {\mathfrak {e}}^{\otimes \lambda _\eta }_{\imath _\eta }\big )}= {{\mathfrak {e}}^{\otimes \lambda _1}_{\imath _1}\odot \ldots \odot {\mathfrak {e}}^{\otimes \lambda _\eta }_{\imath _\eta }}\in H_\eta ^{\odot n}\). Taking all \(\imath \in {\mathscr {I}}\), we conclude that the system indexed by semistandard \(\lambda \)-tabloids
forms an orthogonal basis in the symmetric tensor power \(H_\eta ^{\odot n}\).
The system \(\big \{{\mathfrak {e}}_\imath ^{\otimes \lambda }:= {\mathcal {S}}_n\big ({\mathfrak {e}}^{\otimes \lambda _1}_{\imath _1}\otimes \ldots \otimes {\mathfrak {e}}^{\otimes \lambda _\eta }_{\imath _\eta }\big ):\imath ^\lambda \vdash n, \ \lambda \in {\mathbb {Y}}_n, \ \imath \in {\mathscr {I}}\big \}\), additionally indexed by all \(\sigma \in S_n\), forms an orthonormal basis in the whole tensor power \(H^{\otimes n}\).
As usually, the symmetric Fock space is defined to be the Hilbertian orthogonal sum \(\varGamma (H)={\bigoplus }_{n\ge 0} H^{\odot n}\) with the orthogonal basis \({\mathfrak {e}}^{{\mathbb {Y}}}:={\bigcup \big \{{\mathfrak {e}}^{{\mathbb {Y}}_n}:}{n}\in {\mathbb {N}}_0\big \}\) of elements \(\psi =\bigoplus \psi _n\) with \(\psi _n\in H^{\odot n}\) endowed with the inner product and norm
Note that by tensor multinomial theorem the Fourier expansion under \({\mathfrak {e}}^{{\mathbb {Y}}_n}\)
holds in \(H^{\odot n}\) for all \(h\in H\). Consequently, the linearly independent, so-called, coherent states \({\big \{\exp (h):{h}\in {H}\big \}}\) in \(\varGamma (H)\) have the expansion under the basis \({\mathfrak {e}}^{\mathbb {Y}}\)
with \(h^{\otimes 0}=1\), that is convergent, since \(\Vert {\mathfrak {e}}^{\odot \lambda }_\imath \Vert ^2_\varGamma = n!\Vert {\mathfrak {e}}^{\odot \lambda }_\imath \Vert ^2 \) and
Definition 1
For any \(h\in H\) and \({\mathfrak {u}}\in {\mathfrak {U}}\) the Paley–Wiener maps are defined to be
where projections \(\pi _i:{\mathfrak {U}}\ni {\mathfrak {u}} \rightarrow u_i\in U(i)\) are uniquely defined by \(\pi _i^{i+1}\).
These maps satisfy the orthogonal conditions \(\phi _{{\mathfrak {e}}_i}=\phi _i\) and have the natural extension \({\phi }_{h^*}=\bar{{\phi }_h}\) onto the adjoint space \(H^*\).
Note that, as in the case of linear spaces (see e.g. [12, n.4.4], [29]), the Paley–Wiener maps uniquely determine the embedding \(\phi :H\ni {h}\longmapsto \phi _h\in L^2_\chi \).
For every \({h\in H}\) the \({l}_2\)-valued function \({\phi }_h({\mathfrak {u}})\) of variable \({{\mathfrak {u}}\in {\mathfrak {U}}}\) is well-defined, since \(({\mathfrak {e}}_i^*(h))\in {l}_2\) and \(\left| \left\langle u_i({\mathfrak {e}}_i)\mid {\mathfrak {e}}_i\right\rangle \right| \le 1\). We show that \({\phi }_h\in L^2_\chi \). Assign for any partition \(\lambda =(\lambda _1,\ldots ,\lambda _\eta )\in {\mathbb {N}}^\eta \) of the weight \(|\lambda |={\lambda _1+\ldots +\lambda _\eta }\) the constant
Lemma 2
To every semistandard tableau \([\imath ^\lambda ]\) one can uniquely assign the function
of variable \(u\in {\mathfrak {U}}\) belonging to \(L^\infty _\chi \). The system of \(\chi \)-essentially bounded functions
is orthogonal in the space \(L^2_\chi \) and is normed as follows
Proof
According to (4), we have \((\pi _m\circ \pi _{m+l}^{-1}) u_{m+l}({\mathfrak {e}}_m)= u_m({\mathfrak {e}}_m)\) for \(t=-1\) and \((\pi _m\circ \pi _{m+l}^{-1})u_{m+l}({\mathfrak {e}}_m)=u_m({\mathfrak {e}}_m)-[a(1+t)^{-1}b]{\mathfrak {e}}_m\) for \(t\ne -1\) for any integer \(l\ge 1\). This means that \({({\phi }_k\circ \pi _m^{-1})(u_m)}= \left\langle u_m({\mathfrak {e}}_m) \mid {\mathfrak {e}}_k\right\rangle {\equiv \!\!\!\!\!\!/} 0\) for all \(k\le m\) and that
Let \(U(\eta )\) with \(\eta =\eta (\lambda )\) be the unitary group acting over the linear complex \(\mathop {\text {span}}\left\{ {\mathfrak {e}}_{\imath _1},\ldots ,{\mathfrak {e}}_{\imath _\eta }\right\} \) in H. Let \(\chi _\eta \) be the probability Haar measure on \(U(\eta )\) and \(\pi _\eta :{\mathfrak {U}}\rightarrow U(\eta )\) be the corresponding projector. Using (6) and (17), we obtain
By (18) and the known integral formula for unitary groups \(U(\eta )\) [28, 1.4.9], we get
On the other hand, the invariant property (8) provides the formula
From (19) it follows the orthogonality relations \({\phi ^{\lambda '}_\jmath \perp \phi ^\lambda _\imath }\) with \({|\lambda '|\ne |\lambda |}\), since
for any \(\lambda ',\lambda \in {\mathbb {Y}}{\setminus }\{\emptyset \}\). Let \(|\lambda '|=|\lambda |\) and \(\eta (\lambda ')>\eta (\lambda )\) for definiteness. Then there exists an index k with a nonzero integer \(\lambda '_k\) in \(\lambda '=\big (\lambda '_1,\ldots ,\lambda '_k,\ldots ,\lambda '_{\eta (\lambda ')}\big )\in {\mathbb {Y}}{\setminus }\{\emptyset \}\) such that \(\eta (\lambda )<k\le \eta (\lambda ')\). In this case \({{\phi }^{\lambda '}_\jmath \perp {\phi }^\lambda _\imath }\) because (19) yields
Consider the case \(|\lambda '|=|\lambda |\) and \(\eta (\lambda ')=\eta (\lambda )\). If \({{\phi }^{\lambda '}_\jmath \ne {\phi }^\lambda _\imath }\) then \(\lambda '\ne \lambda \). There exists an index \(0<k\le \eta (\lambda )\) such that \(\lambda '_k\ne \lambda _k\). As above, \({{\phi }^{\lambda '}_\jmath \perp {\phi }^\lambda _\imath }\), because
This proves that the system \({\phi }^{\mathbb {Y}}\) is orthogonal. \(\square \)
4 Orthonormal Basis of Schur Polynomials
Let \(\imath ^\lambda \vdash n\), \(\eta =\eta (\lambda )\) and \(t_\imath =(t_{\imath _1},\ldots ,t_{\imath _\eta })\) be a complex variable. Let \(t^\lambda _\imath :=\prod t_{\imath _j}^{\lambda _j}\). The n-homogenous Schur polynomial is defined (see, e.g. [18]) to be
\(\varDelta (t_\imath )={\prod _{1\le i<j\le \eta }(t_{\imath _i}-t_{\imath _j})}\) is Vandermonde’s determinant. It can be written as \(s^\lambda _\imath (t_\imath )= {\sum }_{[\imath ^\lambda ]}t^\lambda _\imath \) with summation over all semistandard Young tabloids [8, I.2.2].
We construct an orthonormal basis in \(L^2_\chi \) consisting of Schur polynomials on Paley–Wiener maps. Assign (uniquely) to \(\imath \in {\mathscr {I}}_\eta \) the vector \({\phi }_\imath :=\big ({\phi }_{\imath _1},\ldots ,{\phi }_{\imath _\eta }\big )\). Let \(s^\lambda _\imath ({\mathfrak {u}})=(s^\lambda _\imath \circ {\phi }_\imath )({\mathfrak {u}})\) be n-homogeneous functions of variable \({{\mathfrak {u}}\in {\mathfrak {U}}}\) with \(\lambda \in {\mathbb {N}}^\eta \), defined by the formulas (3). Denote
Theorem 1
The system of Schur polynomials \(s^{\mathbb {Y}}\) forms an orthonormal basis in \(L^2_\chi \) and \(s^{\mathbb {Y}}_n\) is the same basis in \(L_\chi ^{2,n}\). The following orthogonal decomposition holds,
For any \({h\in {H}}\) the equality (2) uniquely defines the conjugate-linear embedding
Proof
Let \(U(\eta )\) be the unitary group over the linear complex \(\mathop {\text {span}}\left\{ {\mathfrak {e}}_{\imath _1},\ldots ,{\mathfrak {e}}_{\imath _\eta }\right\} \) with \(\eta =\eta (\lambda )\). Taking into account (17) similarly as (18), we obtain
for all \([\imath ^\lambda ]\), \([\imath ^\mu ]\) with \(\imath =(\imath _1,\ldots ,\imath _\eta )\) and \(\lambda ,\mu \in {\mathbb {N}}^\eta \). In fact, the corresponding Schur polynomials \(\big \{s^\lambda _\imath :\lambda \in {\mathbb {N}}^\eta \big \}\) are characters of the group \(U(\eta )\). Hence, by the Weyl integration formula, the right-hand side integral is equal to Kronecker’s delta \(\delta _{\lambda \mu }\) [26, Thm 8.3.2 & Thm 11.9.1].
The family of finite alphabets \({\imath \in {\mathscr {I}}}\) is directed and for any \(\imath ,\imath '\) there exists \(\imath ''\) such that \(\imath \cup \imath '\subset \imath ''\). This means that the whole system \(s^{\mathbb {Y}}_n\) is orthonormal in \(L^2_\chi \).
The property \({s_\jmath ^\mu \perp s_\imath ^\lambda }\) with \({|\mu |\ne |\lambda |}\) for any \({\imath ,\jmath \in {\mathscr {I}}}\) follows from (19), since
for all \(\lambda \in {\mathbb {Y}}\) and \(\mu \in {\mathbb {Y}}{\setminus }\{\emptyset \}\). This yields \(L_\chi ^{2,|\mu |}\perp L_\chi ^{2,|\lambda |}\) in the space \(L^2_\chi \). Taking \(\lambda =\emptyset \) with \(|\emptyset |=0\), we get \(1\perp L_\chi ^{2,|\mu |}\) for all \(\mu \in {\mathbb {Y}}{\setminus }\{\emptyset \}\). Hence, (20) is proved.
By Lemma 2 the subsystem \(\phi _k=s^1_k\) is orthonormal in \(L^2_\chi \), hence by Definition 1 it instantly follows that \(\Vert {\phi }_h\Vert _\chi ^2=\sum |{\mathfrak {e}}_k^*(h)|^2\int |\phi _k|^2d\chi =\Vert h\Vert ^2.\) It follows the isometric embedding (21).
The set \({\check{U}}(m)\) of matrices with no eigenvalue \(\{-1\}\) has Stone–Ĉech compactification \({\tilde{U}}(m)\) such that the mapping \({\check{\pi }}^{m+1}_m\) has a continuous U(m)-valued extension
This fact follows from [33, Thm 19.5] by virtue of that U(m) is compact. Hence, the projective limit \(\tilde{{\mathfrak {U}}}:=\varprojlim {\tilde{U}}(m),\) determined by \({\tilde{\pi }}^{m+1}_m\), is a compact set in \({\mathfrak {U}}\) with continuous U(m)-valued projections \({{\tilde{\pi }}}_m:\tilde{{\mathfrak {U}}}\rightarrow {U}(m)\).
Since \(U(\infty )\) on H acts irreducibly, for any \({\mathfrak {u}}'\ne {\mathfrak {u}}''\) there is m such that
i.e., \(\phi ^{\mathbb {Y}}\) separates \({\mathfrak {U}}\) and so \(\tilde{{\mathfrak {U}}}\). Hence, the system of Schur polynomials \(s^{\mathbb {Y}}\) also separates \(\tilde{{\mathfrak {U}}}\). Moreover, each complex-conjugate function \({{\bar{\phi }}}_m({\mathfrak {u}})= \left\langle {\mathfrak {e}}_m\mid \pi _m({\mathfrak {u}})\right. \left. ({\mathfrak {e}}_m)\right\rangle = \left\langle \pi _m({\mathfrak {u}}^\star )({\mathfrak {e}}_m)\mid {\mathfrak {e}}_m\right\rangle \) belongs to \(\phi ^{\mathbb {Y}}\). Thus, by the Stone–Weierstrass approximation theorem the complex linear span of polynomials \(\phi ^{\mathbb {Y}}\), as well as, of \(s^{\mathbb {Y}}\), forms a dense subspace in the Banach space of all continuous functions \(C(\tilde{{\mathfrak {U}}})\).
Let \({{\tilde{\chi }}}_m\) means the image of \(\chi _m\) under \({\check{U}}(m)\looparrowright {U}(m)\). In Lemma 1 it inductively was shown that for every \(\varepsilon >0\) there exists a compact set \(\varprojlim K_m\subset \check{{\mathfrak {U}}}\) such that
where \({\tilde{\chi }}_m(K_m)={\check{\chi }}_m(K_m)=\chi _m(K_m)\), by definition of the measure \({{\tilde{\chi }}}_m\) as an image. Hence, by the Prokhorov theorem the projective limit \({\tilde{\chi }}=\varprojlim {\tilde{\chi }}_m\), defined by mappings \({\tilde{\pi }}^{m+1}_m\), possesses the properties
for all Borel \(\varOmega \) in \(\check{{\mathfrak {U}}}\) or otherwise \({{\tilde{\chi }}}|_{\check{{\mathfrak {U}}}}=\chi |_{\check{{\mathfrak {U}}}}\). Consequently,
In particular, \({\tilde{\chi }}=\varprojlim {\tilde{\chi }}_m\) is regular on \(\tilde{{\mathfrak {U}}}\) by the Riesz–Markov theorem [20, 1.1].
As a consequence, the space \(L^2_\chi \) coincides with the completion of \(C(\tilde{{\mathfrak {U}}})\) and for any \({f\in L_\chi ^2}\) there exists a sequence \({(f_n)\subset \mathop {\text {span}}(s^{\mathbb {Y}})}\) such that \({\int |f-f_n|^2d\chi }{\rightarrow 0}\). Hence, the system \(s^{\mathbb {Y}}\) forms an orthogonal basis in \(L_\chi ^2\).
Finally, \(s^{\mathbb {Y}}_n\cap L^2_\chi \) is total in \(L_\chi ^{2,n}\) and \(s^{\mathbb {Y}}_n\perp s^{\mathbb {Y}}_m\) if \(n\ne m\). This yields (20).
\(\square \)
5 Unitarily-Weighted Symmetric Fock Space
Define on the tensor power \({H}^{\otimes n}\) the unitarily-weighted norm \(\Vert \cdot \Vert _{H^{\otimes n}_\beta }= {\langle \cdot \mid \cdot \rangle ^{1/2}_{H^{\otimes n}_\beta }}\) where the inner product \({\langle \cdot \mid \cdot \rangle ^{1/2}_{H^{\otimes n}_\beta }}\) is determined by the relations
Here \({\mathfrak {e}}^{\otimes \lambda }_\imath :=\sigma '({\mathfrak {e}}^{\otimes \lambda _1}_{\imath _1}\otimes \ldots \otimes {\mathfrak {e}}^{\otimes \lambda _\eta }_{\imath _\eta })\) with \(\eta =\eta (\lambda )\) and \(\sigma '\in S_n\) is fixed. Let \({H}^{\otimes n}_\beta \) be the completion of \(\big \{{H}^{\otimes n},\Vert \cdot \Vert _{H^{\otimes n}_\beta }\big \}\). Its closed subspace, defined by the projection
forms an unitarily-weighted symmetric tensor power \({H}^{\odot n}_\beta \subset {H}^{\otimes n}_\beta \) with the inner product determined by relations \(\langle {\mathfrak {e}}^{\odot \lambda }_\imath \mid {\mathfrak {e}}^{\odot \lambda '}_{\imath '}\rangle _{H^{\otimes n}_\beta }= \beta _\lambda \langle {\mathfrak {e}}^{\odot \lambda }_\imath \mid {\mathfrak {e}}^{\odot \lambda '}_{\imath '}\rangle \) or more specific
Definition 2
The unitarily-weighted symmetric Fock space is defined to be the Hilbertian orthogonal sum \({\varGamma _{\!\beta }(H)=\bigoplus _{n\ge 0}{H}_\beta ^{\odot n}}\) of elements \(\psi =\bigoplus \psi _n\), \(\psi _n\in {H}_\beta ^{\odot n}\) with the orthogonal basis \({\mathfrak {e}}^{{\mathbb {Y}}}= \bigcup \big \{{\mathfrak {e}}^{{\mathbb {Y}}_n}:{n}\in {\mathbb {N}}_0\big \}\) and the following inner product and norm
We immediately notice that \(\Vert h\Vert _\beta ^2=\sum |{\mathfrak {e}}_i^*(h)|^2=\Vert h\Vert ^2\) for all \(h=\sum {\mathfrak {e}}_i{\mathfrak {e}}_i^*(h)\in {H}\).
Lemma 3
The set of coherent states \(\left\{ \exp (h):{h}\in {H}\right\} \) is total in \(\varGamma _\beta (H)\) and the expansion (13) is convergent in \(\varGamma _\beta (H)\). The injections
are contractive and dense. The \(\varGamma _\beta (H)\)-valued function \({H\ni {h}\longmapsto \exp (h)}\) is entire analytic. The shift group, defined to be
for \(a,h\in H\), has a unique linear extension \({{\mathcal {T}}_a:\varGamma _\beta (H)\ni \psi \longmapsto {\mathcal {T}}_a\psi \in \varGamma _\beta (H)}\) such that
Proof
Taking into account that \(\beta _\lambda \le 1\), we get the following inequalities
Hence, (12), (13) are convergent in \(\varGamma _\beta (H)\). This implies that \({h\mapsto \exp (h)}\) is analytic and inclusions \(\varGamma (H)\looparrowright \varGamma _\beta (H)\) and \({{H}^{\odot n}\looparrowright {H}_\beta ^{\odot n}}\) are contractive. By the polarization formula (11) their ranges are dense.
Using the binomial formula \({(h+za)^{\otimes n}}= \bigoplus _{m=0}^{n}\left( {\begin{array}{c}n\\ m\end{array}}\right) (za)^{\otimes m}\odot h^{\otimes (n-m)}\), we find
with the orthogonal projector \({\mathcal {S}}_{n/m}\) defined as \(\psi _m\odot \psi _{n-m}={\mathcal {S}}_{n/m}\left( \psi _m\otimes \psi _{n-m}\right) \in {H}^{\odot n}_\beta \) for all \(\psi _m\in {H}^{\odot m}_\beta \) and \(\psi _{n-m}\in {H}^{\odot (n- m)}_\beta \). By orthogonality \(\Vert {\mathcal {S}}_{n/m}\Vert \le 1\).
Applying the expansions (12) to \(a^{\otimes m}\) and \(h^{\otimes (n-m)}\), by (22), we get
with summations over semistandard tableaux \([\imath ^\lambda ],[\jmath ^\mu ]\) and \(\imath ,\jmath \in {\mathscr {I}}\). Let \((\lambda ,\mu )\in {\mathbb {N}}^{\eta (\lambda ,\mu )}\) be the smallest partition of number n with the length \(\eta (\lambda ,\mu )\) containing the partitions \(\lambda \) for m and \(\mu \) for \(n-m\). Then \(\eta (\lambda ,\mu )\ge \max \{\eta (\lambda ),\eta (\mu )\}\) and so
since \(\frac{(\eta -1)!}{(\eta -1+n)!}\) is decreasing in variable \(\eta \). Thus, the following inequality
holds. Using this inequality and that \(\Vert {\mathcal {S}}_{n/m}\Vert \le 1\), we find
Summing with coefficients 1/m!, we get \(\Vert {\mathcal {T}}_{a}\exp (h)\Vert ^2_\beta \le \exp \big (\Vert a\Vert ^2\big )\Vert \exp (h)\Vert ^2_\beta \). This inequality and totality of \(\left\{ \exp (x):h\in {H}\right\} \) in \(\varGamma _\beta (H)\) yield the required inequality (24). It also follows that \(\varGamma _\beta (H)\) is invariant under \({\mathcal {T}}_{a}\) and that the group property (24) holds, since \(\partial _{a+b}=\partial _a+\partial _b\) for all \(a,b\in H\) by linearity.
\(\square \)
Lemma 4
The mapping \(\phi :H\ni {h}\longmapsto \phi _h\in L^2_\chi \), extended onto \({\mathcal {T}}_{a}\exp (h)\) as
has the unique isometric conjugate-linear extension
defined to be \(\langle \varPhi {\mathfrak {e}}^{\odot \lambda }_\imath \mid f\rangle _\chi = \langle {\mathfrak {e}}^{\odot \lambda }_\imath \mid \varPhi ^*f\rangle _\beta \) for all \({f\in L_\chi ^2 }\) in such way that
As a result, the conjugate-linear isometries \(\varGamma _\beta (H) {\mathop {\simeq }\limits ^{\varPhi }}L^2_\chi \) and \({H}^{\odot n}_\beta {\mathop {\simeq }\limits ^{\varPhi }}L^{2,n}_\chi \) hold.
Proof
By Lemma 3 the \(\varGamma _\beta (H)\)-valued function \(H\ni h\mapsto {\mathcal {T}}_{a}\exp (h)\) is well defined for all \({a\in H}\). Let us use the expansion \(\phi _{h+a}={\sum {\mathfrak {e}}^*_i(h+a)\phi _i}\). By Lemma 2 and Theorem 1, \(\phi :H\ni {h}\longmapsto \phi _h\in L_\chi ^2 \) may be extended to \(\varPhi \) in following way
is an orthogonal component of \(\varPhi {\mathcal {T}}_{a}\exp (h)\) in \(L^2_\chi \). It follows that
Hence, the composition \({\mathfrak {U}}\ni {\mathfrak {u}}\longmapsto [\varPhi \exp (h+a)]({\mathfrak {u}})\) is well defined in \(L^2_\chi \).
Now, we consider the ordinary irreducible representation of permutation group \(S_n\) on the Specht \(\lambda \)-module \(S^\lambda _\imath \) that is corresponded to the standard Young tableau \([\imath ^\lambda ]\). The following known hook formula (see [8, I.4.3]) holds,
with \(h(i,j)\!=\!\#\big \{\Box _{i'j'}\in [\imath ^\lambda ]:i'\ge i,j'=j\big \}\!=\!\#\big \{\Box _{i'j'}\in [\imath ^\lambda ]:{i'=i, j'\ge j}\big \}\) independed of \(\imath \in {\mathscr {I}}\). Assign to \(\imath \in {\mathscr {I}}_\eta \) the vectors
Let \(s^\lambda _\imath ({\mathfrak {u}},h):=s^\lambda _\imath (t_\imath )\) with \(t_\imath =t_\imath ({\mathfrak {u}},h)\) for all \({{\mathfrak {u}}\in {\mathfrak {U}}}\), where polynomial terms are \({\phi }^\lambda _\imath ({\mathfrak {u}}){{\mathfrak {e}}}_\imath ^{*\lambda }(h)= {\phi }^{\lambda _1}_{\imath _1}({\mathfrak {u}}){{\mathfrak {e}}}^{*\lambda _1}_{\imath _1}(h) \ldots {\phi }^{\lambda _\eta }_{\imath _\eta }({\mathfrak {u}}) {{\mathfrak {e}}}^{*\lambda _\eta }_{\imath _\eta }(h)\). Applying the Frobenius formula [18, I.7] and taking into account (2), (3), (25), we obtain
where \(s^\lambda _\imath =0\) if \(\lambda ^\intercal _1>l_\lambda \) and the summation is over all standard tabloids. Hence, \(\big \{\phi _h^n:h\in H\big \}\) is total in \(L_\chi ^{2,n}\) by Theorem 1. In consequence, \(\left\{ \exp (\phi _h):h\in H\right\} \) is total in \(L_\chi ^2 \). This yields surjectivity of \(\varPhi \) and of all its restrictions to \({H}^{\odot n}_\beta \).
\(\square \)
Corollary 2
The sets \(\big \{\phi _h^n:{h}\in {H}\big \}\) in \(L^{2,n}_\chi \) and \(\left\{ \exp \phi _h:{h}\in {H}\right\} \) in \(L^{2}_\chi \) are total.
6 Fourier Analysis on Virtual Unitary Matrices
Consider the isometry \({H}^{*\odot n}_\beta {\mathop {\simeq }\limits ^{{\mathcal {P}}}}{P}_\beta ^n(H)\) (see e.g., [7, 1.6]), where the space \({P}_\beta ^n(H)\) of unitarily-weighted n-homogeneous Hilbert–Schmidt polynomials of variable \(h\in H\) is defined to be a restriction to the diagonal in \({H\times \ldots \times H}\) of the n-linear forms \({{\mathcal {P}}\circ \psi _n}\) endowed with the norm \(\Vert \psi _n^*\Vert _{P_\beta ^n}=\Vert \psi _n\Vert _{H^{\otimes n}_\beta }\) where
Let \({H}^2_\beta =\sum _{n\ge 0}{P}_\beta ^n(H)\) be the direct sum of functions \(\psi ^*(h)=\sum \psi _n^*(h)\) of variable \({h\in H}\) with summands \(\psi _n^*={\mathcal {P}}\circ \psi _n\in {P}_\beta ^n(H)\) where \(\psi =\sum \psi _n\in \varGamma _\beta (H)\). Since the set \(\{\exp (h):h\in H\}\) is total in \(\varGamma _\beta (H)\), elements of \({H}^2_\beta \) can be written as
The analyticity of \(H\ni {h}\mapsto \psi ^*(h)\) is a result of the composition \(\exp (\cdot )\) and \(\psi ^*(\cdot )\).
Definition 3
Let \({H}^2_\beta \) be defined as a Hardy space of unitarily-weighted Hilbert–Schmidt analytic functions \(\psi ^*(h)\) of variable \({h\in H}\) endowed with the inner product
The conjugate-linear surjective isometry from \({H}^2_\beta \) onto \(\varGamma _\beta (H)\) is realized by the conjugate-linear mapping
On the other hand, the correspondence \(\varPhi :{\mathfrak {e}}^{\odot \lambda }_\imath \rightleftarrows \phi _\imath ^\lambda \) with \(\lambda \in {\mathbb {Y}}\) and \(\imath \in {\mathscr {I}}_{\eta (\lambda )}\) allows us to determine the conjugate-linear isometry from \(\varGamma _\beta (H)\) onto \(L^2_\chi \). As a result, the mapping
defines the surjective isometry
Lemma 5
The systems of Hilbert–Schmidt polynomials of variable \({h\in {H}}\),
where \({{\mathfrak {e}}^{*\emptyset }_\imath =1}\), form orthogonal bases in \({P}_\beta ^n(H)\) and \({H}^2_\beta \), respectively, such that
Every function \({\psi ^*\in {H}^2_\beta }\) with \(\psi \in \varGamma _\beta (H)\) has the expansion with respect to \({\mathfrak {e}}^{*{\mathbb {Y}}}\)
with summation in the inner sum over all semistandard tabloids \({[\imath ^\lambda ]}\) such that \({\imath ^\lambda \vdash n}\). Each function \({\psi ^*\in {H}^2_\beta }\) is entire Hilbert–Schmidt analytic and can be also written as
The following linear isometries, defined by linearization via coherent states, hold
Proof
Taking into account (13) and (23), we conclude that every \(\psi ^*\in {H}^2_\beta \) such that \(\psi =\bigoplus \psi _n\in \varGamma _\beta (H)\) with \({\psi _n\in {H}^{\odot n}_\beta }\) has the following expansion
On the other hand, in relative to the inner product \(\langle \cdot \mid \cdot \rangle _\varGamma \), we have
Verify the first equality in (27) by substituting (26) into the formula (27). We get
If \({\omega ^*(h'):=\psi ^*(h)\exp \langle h\mid h'\rangle [\exp \langle h'\mid h'\rangle ]^{-1}}\) then \(\omega ^*(h)=\psi ^*(h)\) for \({h=h'\in {H}}\). Now, putting \(\omega ^*(h'):= \big \langle \psi ^*(\cdot )\mid \exp \langle h'\mid \cdot \rangle [\exp \langle h'\mid h'\rangle ]^{-1} \exp \langle \cdot \mid h'\rangle \big \rangle _{H^2_\beta }\), we obtain
Hence, the second equality in (27) holds. Lemma 4 yields (28). \(\square \)
Remark 1
Since \(\phi _h=\sum {\mathfrak {e}}^*_i(h)\phi _i\) for all \(h=\sum {\mathfrak {e}}^*_i(h){\mathfrak {e}}_i\), a range of the embedding (21) coincides with \(L_\chi ^{2,1}\).
Lemma 6
Denote \(\exp \langle h'\mid h\rangle :=K(h',h)\). The functions
with \({\mathfrak {u}}\in {\mathfrak {U}}\) take values in \(L^2_\chi \) and can be represented as follows
where the last exponential function has the power series expansion
with coefficients in the form of complex Hermite polynomials \({\mathfrak {h}}_{n,m}(z,{\bar{z}})\), \(z\in {\mathbb {C}}\).
Proof
Applying the transform \(\varPsi \) to \({K}(h',h)\) in variable \({h'\in {H}}\), we obtain
Similarly, applying \(\varPsi \) to \({E}(h',h)\) in variable \({h'\in {H}}\), we obtain
By Lemma 4, \((\varPsi \circ {K})(\cdot ,h)\) and \((\varPsi \circ {E})(\cdot ,h)\) with \({h\in {H}}\) take values in \( L_\chi ^2 \). The expansion (29) follows from [13, n.12] where polynomials \({\mathfrak {h}}_{n,m}(z,{\bar{z}})\) were introduced. \(\square \)
Theorem 2
For any \(f={\sum f_n\in L^2_\chi }\) with \(f_n\in L^{2,n}_\chi \) the entire function
and its Taylor coefficients at zero \(d^n_0{\hat{f}}\) have the integral representations
respectively. The Fourier transform \({F:L^2_\chi \ni {f}\longmapsto {\hat{f}}\in {H}^2_\beta }\) provides the isometries
Proof
Since \(\varPsi =\varPhi \circ *^{-1}\), we obtain \(\varPsi ^*=*\circ \varPhi ^*\). From (27) it follows that \({\hat{f}}(h)=\left\langle \exp (h)\mid \varPhi ^*f\right\rangle _\beta = {\big \langle (\varPsi ^*\circ f)(\cdot )\mid {K}(\cdot ,h)\big \rangle _{{H}^2_\beta }} =\big \langle (\varPsi ^*\circ f)(\cdot )\mid {E}(\cdot ,h)\big \rangle _{{H}^2_\beta }\). Thus,
by Lemma 6. On the other hand, according to the same claim
It particularly follows that for all \(h=\alpha {x}\) with \(x\in H\),
Using the n-homogeneity of derivatives, we find
Finally, we notice that the isometry \({L^2_\chi {\mathop {\simeq }\limits ^{F}}{H}^2_\beta }\) holds, since the isometry \(\varPhi ^*\) is surjective by Lemma 5. Similarly, we get \(L^{2,n}_\chi {\mathop {\simeq }\limits ^{F}}{P}_\beta ^n(H)\). \(\square \)
Corollary 3
For any \(h\in {H}\) the Paley–Wiener map \(\phi _h\) satisfies the equality
Proof
It is enough to put \(f\equiv 1\) and to replace h by h/2 in the formula (30).
\(\square \)
Corollary 4
The isometry \({*:\varGamma _\beta (H)\longrightarrow {H}^2_\beta }\) has the factorization \(*={F}\circ \varPhi \).
Proof
In fact, \({\varPhi :\varGamma _\beta (H)\ni \psi \longmapsto \varPhi \psi =f\in L^2_\chi }\) and \({F}:L^2_\chi \ni f\longmapsto {\hat{f}}\in {H}^2_\beta \).
\(\square \)
Corollary 5
For every \(f\in L^2_\chi \) the Taylor expansion at zero of the function
has the coefficients
with summation over all standard Young tabloids \({[\imath ^\lambda ]}\) such that \({\imath ^\lambda \vdash n}\) where \(s^\lambda _\imath =0\) if the conjugate partition \(\lambda ^\intercal \) has \(\lambda ^\intercal _1>\eta (\lambda )\) and \({s}^\lambda _\imath [{f}_\imath \,{{\mathfrak {e}}}_\imath ^*(h)]:= s^\lambda _\imath (t_\imath )\) with \(t_\imath ={f}_\imath \,{{\mathfrak {e}}}_\imath ^*(h)\).
Proof
By the Frobenius formula [18, I.7] we find that \(\phi _h^n({\mathfrak {u}})= \sum _{\imath ^\lambda \vdash n}\hbar _\lambda s^\lambda _\imath ({\mathfrak {u}},h)\), where \(s^\lambda _\imath =0\) if \(\lambda ^\intercal _1>\eta (\lambda )\), and \(s^\lambda _\imath ({\mathfrak {u}},h)\) is defined by (3), whereas \(\hbar _\lambda \) by (25). Thus,
Using (32) in combination with Theorem 1, we find
where the derivative at zero may be defined as
In fact, for zh with \({z\in {\mathbb {C}}}\) and \(\imath ^\lambda \vdash n\) with \(\lambda ^\intercal _1>\eta (\lambda )\) we find
Hence, the derivative \(d^n_0{\hat{f}}(h)=(d^n/dz^n){\hat{f}}(zh)|_{z=0}\) is a Taylor coefficient of \({\hat{f}}\).
Now, the Frobenius formula and Theorem 1 yield the first equality in (31). By Lemmas 5 and 6 the second formula in (31) also holds. \(\square \)
Remark 2
In the finite-dimensional case \({\mathfrak {U}}=U(m)\), the Hardy space \({H}^2_\beta \) of entire analytic functions of variable \(h\in {\mathbb {C}}^m\) has the following orthogonal basis \(\big \{{\mathfrak {e}}^{*\lambda }={\mathfrak {e}}^{*\lambda _1}_1\ldots {\mathfrak {e}}^{*\lambda _m}_m :{\lambda =(\lambda _1,\ldots ,\lambda _m)\in {\mathbb {Y}}}\big \}\). The Fourier transform
provides the surjective isometry \({F:L_{\chi _m}^2\ni {f}\longmapsto {\hat{f}}\in {H}^2_\beta }\), defined by mappings
where the space \(L_{\chi _m}^2\) with the Haar measure \(\chi _m\) on U(m) has the orthogonal basis \(\big \{\phi ^\lambda = {\phi }^{\lambda _1}_1\circ \pi _m^{-1}\ldots {\phi }^{\lambda _m}_m\circ \pi _m^{-1}:\lambda \in {\mathbb {Y}}\big \}\).
7 Intertwining Properties of Fourier Transform
The shift group on \({H}^2_\beta \) is defined as \(T_a\psi ^*(h):=\left\langle {\mathcal {T}}_a\exp (h)\mid \psi \right\rangle _\beta \) for all \(\psi \in \varGamma _\beta (H)\), \({a,h\in H}\). By (27), \(\left\langle {\mathcal {T}}_a\exp (h)\mid \psi \right\rangle _\beta =T_a\psi ^*(h)=\big \langle T_a\psi ^*(\cdot )\mid \exp \langle \cdot \mid h\rangle \big \rangle _{H^2_\beta }\). Hence,
where \({M}_{a^*}\exp \langle \cdot \mid h\rangle :=\exp {a}^*(\cdot )\exp \langle \cdot \mid h\rangle =\exp \langle \cdot \mid h+a\rangle \) is defined to be the multiplicative group onto the total set \(\{{\exp \langle \cdot \mid h\rangle }:h\in H\}\) in \({H}^2_\beta \).
Comparing the above formulas, we obtain that \({M}_{a^*}\) is adjoint to \({T}_a\) on \({H}^2_\beta \). By virtue of adjoint relations, \(\Vert {T}_a\psi ^*\Vert _{H^2_\beta } =\Vert {M}_{a^*}\psi ^*\Vert _{H^2_\beta }\). The isometry \(H^2_\beta \simeq \varGamma _\beta (H)\) yields \(\Vert {T}_a\psi ^*\Vert _{H^2_\beta }=\Vert {\mathcal {T}}_a\psi \Vert _\beta \). According to (24), we have
for \({a,b\in H}\). Thus, these groups are strongly continuous with densely defined closed generators \(\partial ^*_a\psi ^*:={\lim _{z\rightarrow 0}(T_{za}\psi ^*-\psi ^*)/z}\) and \({a}^*\psi ^*:={\lim _{z\rightarrow 0}(M_{za^*}\psi ^*}{-\psi ^*)/z}\).
Hence, the additive group \((H,+)\) on \({H}^2_\beta \) is represented by \({M}_{a^{*}}:{H}^2_\beta \rightarrow {H}^2_\beta \) and the generator \(dM_{za^{*}}/dz\mid _{z=0}=a^{*}\) of its 1-parameter subgroup \(M_{za^{*}}\) is strongly continuous with the dense domain \({\mathfrak {D}}(a^{\!*})={\big \{\psi ^*\in {H}^2_\beta :a^{*}\psi ^*\in {H}^2_\beta \big \}}\). On the other hand, the group \((H,+)\) can be represented as \(M_{a^{*}}^\dagger =\varPsi {M}_{a^{*}}\varPsi ^* :L^2_\chi \rightarrow L^2_\chi \). The generator of its strongly continuous subgroup
has the dense domain \({\mathfrak {D}}({{\bar{\phi }}}_a)={\big \{f\in L^2_\chi :{{\bar{\phi }}}_a f\in L^2_\chi \big \}}\) and is closed, since \(a^*\) is closed.
The group \((H,+)\) on \(L^2_\chi \) can be also represented by \(T_a^\dagger :=\varPsi {T}_a\varPsi ^*:L^2_\chi \rightarrow L^2_\chi \). From Lemmas 3 and 5 it follows that the generator of strongly continuous subgroup
has the dense domain \({\mathfrak {D}}(\partial _a^\dagger )={\big \{f\in L^2_\chi :\partial _a^\dagger f\in L^2_\chi \big \}}\) and is closed, since \(\partial _a^*\) is closed. By (27) \({\hat{f}}(h)=\left\langle \exp (h)\mid \varPhi ^*f\right\rangle _\beta = \big \langle (\varPsi ^*\circ f)(\cdot )\mid \exp \langle \cdot \mid h\rangle \big \rangle _{H^2_\beta }\). Hence, by Lemma 6,
Lemma 7
The additive group \((H,+)\) on \(L^2_\chi \) has two representations \({a\mapsto M_{a^{*}}^\dagger }\) and \({a\mapsto T_a^\dagger }\) which are adjoint, strongly continuous with closed densely defined generators \({{\bar{\phi }}}_a\) and \(\partial _a^\dagger \), respectively. For every \(f\in {\mathfrak {D}}({{\bar{\phi }}}_a^m)={\big \{f\in L^2_\chi :{{\bar{\phi }}}_a^m{f}}{\in L^2_\chi \big \}}\) with \(m\in {\mathbb {N}}_0\),
For every \(f\in {\mathfrak {D}}(\partial _a^{\dagger m})={\big \{f\in L^2_\chi :\partial _a^{\dagger m}{f}\in L^2_\chi \big \}}\) with \(m\in {\mathbb {N}}_0\),
As a conclusion, \(\partial _{\mathbb {i}a}^\dagger =-\mathbb {i}\partial _a^\dagger \). Moreover, the following commutation relations hold,
for all f from the dense subspace \({\mathfrak {D}}({{\bar{\phi }}}_a^2)\cap {\mathfrak {D}}(\partial _b^{\dagger 2})\subset {L}^2_\chi \) and nonzero \(a,b\in {H}\).
Proof
Using that \({T}_a\) and \({M}_{a^{\!*}}\) are adjoint, we find that
for all \({f\in L^2_\chi }\). This gives (34). Since \(M_{a^{*}}\psi ^*(h)=\big \langle \psi ^*(\cdot )\mid {M}_{a^{*}}\exp \langle \cdot \mid h\rangle \big \rangle _{H^2_\beta }=\exp {a^{*}}(h)\,\psi ^*(h)\), we obtain
This together with the group property by applying F and \({F}^{-1}\) yields (35).
Now, we prove the commutation relations. For any \({f\in L^2_\chi }\) and \(h\in {H}\), we have
For each \({\hat{f}}\in {\mathfrak {D}}(b^{*2})\cap {\mathfrak {D}}(\partial _a^{2})\) and \({t\in {\mathbb {C}}}\) by differentiation, we obtain
Subsequently, taking into account (38) together with \({(d/dt)[\exp \langle ta\mid {\bar{t}}b\rangle M_{tb^{*}}T_{ta}]}\)\(=[(d/dt)\exp \langle {ta}\mid {\bar{t}}b\rangle ] M_{tb^{*}}T_{ta}+ \exp \langle {ta}\mid {{\bar{t}}b}\rangle [(d/dt)M_{tb^{*}}T_{ta}],\) we find
Hence, for each \({\hat{f}}\) from the dense subspace \({\mathfrak {D}}(b^{*2})\cap {\mathfrak {D}}(\partial _a^{2}) \subset {H}^2_\beta \), which includes all polynomials generated by finite sums \(\varPsi ^*(f)=\bigoplus \psi _n\in \varGamma _\beta (H)\) with \(\psi _n\in {H}^{\odot {n}}_\beta \),
Corollary 4 yields \({F}=*\circ \varPhi ^*\) and \({F}^{-1}=\varPhi \circ *^{-1}\). The equality (37) for \(m=0\) can be rewritten as \(M_{b^{*}}{\hat{f}}(a)= \left\langle \exp (a)\mid {T}_b\varPhi ^* f\right\rangle _\beta \) with \({f\in L^2_\chi }\) or in another way \(*\circ {T}_b=M_{b^{*}}\circ *\). Hence, \(T_b^\dagger =\varPhi \,{T}_b\varPhi ^* =\varPhi \circ {*}^{-1}\circ {M}_{b^*}\circ *\circ \varPhi ^*= {F}^{-1}M_{b^{*}}\,{F}\) and \(\partial _b^\dagger ={F}^{-1}b^{*}\,{F}\). Similarly, \(M_{a^*}^\dagger ={F}^{-1}T_a\,{F}\) and \({{\bar{\phi }}}_a={F}^{-1}\partial _a^*\,{F}\). Finally,
for all f from the dense subspace \({\mathfrak {D}}({{\bar{\phi }}}_a^2)\cap {\mathfrak {D}}(\partial _b^{\dagger 2})\subset {L}^2_\chi \), which includes all functions generated by finite sums \(\varPhi \left( \bigoplus \psi _n\right) \) with \(\psi _n\in {H}^{\odot {n}}_\beta \). \(\square \)
8 Infinite-Dimensional Heisenberg Group
Our goal is to describe an irreducible representation on the space \(L^2_\chi \) of the group \({\mathcal {H}}_{\mathbb {C}}\), defined by (1). We will use the appropriate generalization of Weyl’s system which in our case is written in the form of \(L^2_\chi \)-valued function of variable \({h\in H}\)
For convenience, we will use the quaternion algebra \(\mathbb {H}={\mathbb {C}}\oplus {\mathbb {C}}\mathbb {j}\) of numbers \(\zeta ={{(\alpha _1+\alpha _2\mathbb {i})}+{(\alpha '_1+\alpha '_2\mathbb {i})}\mathbb {j}} ={\alpha +\alpha '\mathbb {j}}\) such that \(\mathbb {i}^2=\mathbb {j}^2=\mathbb {k}^2= \mathbb {i}\mathbb {j}\mathbb {k}=-1\), \(\mathbb {k}=\mathbb {i}\mathbb {j}=-\mathbb {j}\mathbb {i}\), \(\mathbb {k}\mathbb {i} = -\mathbb {i}\mathbb {k}= \mathbb {j}\), where \((\alpha ,\alpha ')\in {\mathbb {C}}^2\) with \(\alpha ={\alpha _1+\alpha _2\mathbb {i}},\alpha '={\alpha '_1+\alpha '_2\mathbb {i}\in {\mathbb {C}}}\) and \(\alpha _\imath ,\alpha '_\imath \in {\mathbb {R}}\)\((\imath =1,2)\) [26, 5.5.2]. Let us denote \(\alpha ':=\mathfrak {I}{\zeta }\) for all \(\zeta =\alpha +\alpha '\mathbb {j}\in \mathbb {H}\).
Consider the Hilbert space \({H}\oplus {H}\mathbb {j}\) with \(\mathbb {H}\)-valued inner product
where \({h}={a}+{b}\mathbb {j}\) with a, \({b}\in {H}\). Hence,
Theorem 3
The representation of \({\mathcal {H}}_\mathbb {C}\) over \(L^2_\chi \) in the Weyl–Schrödinger form
is well defined and irreducible. The Weyl system satisfies the relation
which on any real subspace \(\{\tau {h}:\tau \in {\mathbb {R}}\}\) transforms to the 1-parameter group
with the densely defined generator on \(L^2_\chi \) of the form \({\mathfrak {p}}^\dagger _{h}:=\partial _{b}^\dagger +{{\bar{\phi }}}_{a}\). Moreover, the following commutation relations hold,
on the dense subspace \({\mathfrak {D}}({{\bar{\phi }}}_a^2)\cap {\mathfrak {D}}(\partial _b^{\dagger 2})\subset {L}^2_\chi \).
Proof
Let us consider the auxiliary group \({\mathbb {C}}\times (H\oplus {H}\mathbb {j})\) with multiplication \((t,{h})(t',{h}')= \left( t+t'-\frac{1}{2}\mathfrak {I}\langle {h}\mid {h}'\rangle ,\,{h}+{h}'\right) \) for all \({h}={a}+{b}\mathbb {j}\), \({h}'={a}'+{b}'\mathbb {j}\in {H} \oplus {H} \mathbb {j}\). The mapping \({G}:X({a},{b},t)\longmapsto \left( t-\frac{1}{2}\langle {a}\mid {b}\rangle , \,{a}+{b}\mathbb {j}\right) \) is a group isomorphism, since
On the other hand, let us define the auxiliary Weyl system
Using group properties and the commutation relation (39), we obtain
Hence, the mapping \({{\mathbb {C}}\times ({H} \oplus {H} \mathbb {j})\ni (t,{h})\longmapsto \exp (t){W}({h})}\) acts as a group isomorphism into the operator algebra over \({H}^2_\beta \). So, the representation
is also well defined over \({H}^2_\beta \), as a composition of group isomorphisms.
Let us check the irreducibility. Suppose the contrary. Assume there exist an element \({h}_0\ne 0\) in H and an integer \({n>0}\) such that
But, this is only possible for \({h}_0=0\). It gives a contradiction. Finally, using that
we obtain that \(S^\dagger ={F}^{-1}S\,{F}\) is irreducible. Applying F, \({F}^{-1}\) to (44) we get (40).
Consider the Weyl system \({W}^\dagger \) on the space \(L^2_\chi \). By (40) we obtain the equality
Using this equality, we get (41) for any fixed \({h}={a}+{b}\mathbb {j}\in {H}\oplus {H}\mathbb {j}\). The 1-parameter group \({W}^\dagger (\tau {a},\tau {b})={W}^\dagger (\tau {h})\) with real \(\tau \) has the generator \({\mathfrak {p}}_{h}^\dagger ={\mathfrak {p}}_{a,b}^\dagger \), since
Taking into account the inequalities (33) and that F is isometric, we get
Hence, the group \({W}^\dagger (\tau {a},\tau {b})\) in variable \(\tau \in {\mathbb {R}}\) is strongly continuous on \(L^2_\chi \) and therefore has the dense domain \({\mathfrak {D}}({\mathfrak {p}}_{h}^\dagger ) ={\big \{f\in L^2_\chi :{\mathfrak {p}}^\dagger _{h}f\in L^2_\chi \big \}}\). Moreover, its generator \({\mathfrak {p}}^\dagger _{h}\) is closed (see, e.g., [32]). Note also that \({\mathfrak {p}}^\dagger _{\tau {h}}=\tau {\mathfrak {p}}^\dagger _{h}\) for \(\tau \in {\mathbb {R}}\).
Finally, applying the commutation relation (36) and commutability of group generators in different directions over the dense set \({\mathfrak {D}}({{\bar{\phi }}}_a^2)\cap {\mathfrak {D}}(\partial _b^{\dagger 2})\subset {L}^2_\chi \), we have
\(\square \)
9 Heat Equation Associated with Weyl System
In what follows, we will consider the real Banach space \(c_0\) and let \(\xi _n^*\) be the coordinate functional, i.e., \(\xi _n^*(\xi )=\xi _n\) for \({\xi \in c_0}\). Since, the embedding \({\mathcal {I}}:{l}_2\looparrowright c_0\) is continuous, the Gelfand triple \(l_1 {\mathop {\longrightarrow }\limits ^{{\mathcal {I}}^*}}l_2 \looparrowright c_0\) with adjoint \({\mathcal {I}}^*\) holds. The mapping \(Q:{l}_1\rightarrow c_0\) with \(Q:={\mathcal {I}}\circ {\mathcal {I}}^*\) is positive and \(\langle Q\xi ^*\mid Q\xi ^*\rangle _{l_2}:=\xi ^*(Q\xi ^*)=\sum \xi _n^2=\Vert \xi \Vert ^2_{l_2}\) where \(\xi =Q\xi ^*\in {\mathscr {R}}(Q)\) and \(\xi ^*\in {l}_1=c^*_0\). By the Aronszajn-Kolmogorov decomposition theorem (see e.g., [22, Prop.1]) the appropriative reproducing kernel Hilbert space can be determined as \(\overline{{\mathscr {R}}(Q)}=l_2\).
Consider the abstract Wiener space defined by \({\mathcal {I}}:l_2\looparrowright c_0\). Given \(\xi _1^*,\ldots ,\xi _n^*\in l^1=c_0^*\), we assign the family of cylinder sets \(\varOmega _n^c=\left\{ \xi \in c_0:(\xi _1^*(\xi ),\ldots ,\right. \left. \xi _n^*(\xi ))\in \varOmega _n\right\} \) with any Borel \(\varOmega _n\subset {\mathbb {R}}^n\) that are not a \(\sigma \)-field. Define the \(\sigma \)-additive extension \({\mathfrak {w}}\) of the Gaussian measure \(\gamma \) onto the Borel \(\sigma \)-algebra \({\mathscr {B}}(c_0)\), called futhure the Wiener measure, such that
By Gross’ theorem [10] there exists a smaller abstract Wiener space \(\{w_0,\Vert \cdot \Vert _{w_0}\}\) such that injections \({l_2\looparrowright {w}_0\looparrowright c_0}\) are continuous and the increasing sequence of orthogonal projectors \(p_n:{l}_2\rightarrow {\mathbb {R}}^n\) has the extension \((p^\sim _n)\) on \(w_0\) that is convergent to the identity operator on \(w_0\) and \({{\mathfrak {w}}(w_0)=1}\). The integral of any cylinder function \({\upsilon :c_0\rightarrow {\mathbb {R}}}\) such that \(\upsilon =\rho \circ p_n^\sim \) is defined to be \(\int _{\varOmega _n^c}\upsilon \,d{\mathfrak {w}}=\int _{\varOmega _n}\rho \,d\gamma \). The Fernique theorem [6, 15, Thm 3.1] implies that these exist \(\varepsilon ,\eta >0\) such that \(\Vert \cdot \Vert _{w_0}\) satisfies the following conditions with a sufficiently large \(K>0\),
Let us go back to the Weyl system \({W}^\dagger \). Consider in \(L^2_\chi \) the dense subspace \({L}^{+2}_\chi :=\bigcup _{n\ge 0}\bigoplus _{m=0}^nL^{2,m}_\chi \). Let \({a}={b}=\mathbb {i}\xi _m{\mathfrak {e}}_m\) with \({\xi _m\in {\mathbb {R}}}\). Then by Theorem 3
Theorem 4
For any \(f\in {L}^{+2}_\chi \) and \(\xi =(\xi _m)\in c_0\) there exists the limit
\({\mathfrak {w}}\)-almost everywhere on \(c_0\) such that the 1-parameter Gaussian semigroup
on the space \({L}^{+2}_\chi \) is generated by \(-\sum \big (\partial _m^\dagger +{{\bar{\phi }}}_m\big )^2\) with \(\partial _m^\dagger :=\partial _{{\mathfrak {e}}_m}^\dagger \). As a consequence, \(w(r)={\mathfrak {G}}^\dagger _rf\) is unique solution of the Cauchy problem
Proof
Note that \((M_{b^*}T_a)^*=T_a^*M_{b^*}^*=M_{a^*}T_b\). Hence, \((\partial _{a}^\dagger +{{\bar{\phi }}}_{a})^*=\partial _{a}^\dagger +{{\bar{\phi }}}_{a}\) is self-adjoint for \(a=b\), as a generator of the group \({W}^\dagger (\tau {a},\tau {a})=\exp \left\{ {\Vert \tau a\Vert ^2}/{2}\right\} T_{\tau {a}}^\dagger M_{\tau {a}^*}^\dagger \) with \({\tau \in {\mathbb {R}}}\). Replacing \(a=b\) by \(\mathbb {i}\tau a\) with \({\tau \in {\mathbb {R}}}\), we obtain that
with self-adjoint \(\partial _{a}^\dagger +{{\bar{\phi }}}_{a}\). By relations (36), \({W}^\dagger (\mathbb {i}\tau {a},\mathbb {i}\tau {a})\) is unitary.
Lemma 7 implies that \([M_{-\mathbb {i}\xi _m{\mathfrak {e}}_m^*}^\dagger , T_{\mathbb {i}\xi _k{\mathfrak {e}}_k}^\dagger ]=0\) and \({[M_{-\mathbb {i}\xi _m{\mathfrak {e}}_m^*}^\dagger , M_{-\mathbb {i}\xi _k{\mathfrak {e}}_k^*}^\dagger ]=0}\), as well as, \([T_{\mathbb {i}\xi _m{\mathfrak {e}}_m}^\dagger , T_{\mathbb {i}\xi _k{\mathfrak {e}}_k}^\dagger ]=0\) for any \(m\ne k\). In view of the relations (36),
Check that (45) holds. Denote \({W}^\dagger _{p_n^\sim (\xi )}:= \prod _{m=1}^{n}{W}^\dagger (\mathbb {i}\xi _m{\mathfrak {e}}_m,\mathbb {i}\xi _m{\mathfrak {e}}_m)\) and \(T^\dagger _{p_n^\sim (\xi )}:=\prod _{m=1}^{n} T^\dagger _{\mathbb {i}\xi _m{\mathfrak {e}}_m}\), as well as, \(M^\dagger _{p_n^\sim (\xi )}:=\prod _{m=1}^{n} M^\dagger _{-\mathbb {i}\xi _m{\mathfrak {e}}_m^*}\) with \({\xi =(\xi _m)}{\in {w}_0}\). Using (33) with the operator norm over \(H^2_\beta \), we get the inequality
The relation \(T_{\mathbb {i}\xi _m{\mathfrak {e}}_m}^\dagger =\varPsi {T}_{\mathbb {i}\xi _m{\mathfrak {e}}_m}\varPsi ^*\) implies that the left-hand side term above can be changed by \(\ln \prod _{m=1}^{n}\Vert T_{\mathbb {i}\xi _m{\mathfrak {e}}_m}^\dagger \Vert _{{\mathscr {L}}({L}^2_\chi )}^2\). For \(M^\dagger _{p_n^\sim (\xi )}=\prod _{m=1}^{n} M^\dagger _{-\mathbb {i}\xi _m{\mathfrak {e}}_m^*}\) similarly.
Using the unitarity of groups \({W}^\dagger (\mathbb {i}\xi _m{\mathfrak {e}}_m,\mathbb {i}\xi _m{\mathfrak {e}}_m)\), we find by virtue of (47) that their product \({W}^\dagger _{p_n^\sim (\xi )}= \exp \left\{ -\Vert {p_n^\sim (\xi )}\Vert _{l_2}^2/2\right\} T^\dagger _{p_n^\sim (\xi )} M^\dagger _{p_n^\sim (\xi )}\) is also unitary. Taking into account the continuity of \({{\mathcal {I}}_0:l_2\looparrowright {w}_0}\) and that \(p_n^\sim \) converges to the identity mapping on \(w_0\), as well as, that \({{\mathfrak {w}}(w_0)=1}\), we obtain for all \({f\in {L}^{+2}_\chi }\), \(n\ge 0\),
The Lebesgue dominated convergence theorem implies that there exists \(\lim \Vert {W}^\dagger _{p_n^\sim (\xi )}f\Vert _\chi \)\({\mathfrak {w}}\)-almost everywhere in variable \({\xi \in {w}_0}\) for all \({f\in L^{2,m}_\chi }\) and \({m>0}\). By completeness of \(L^{2,m}_\chi \), the limit \({W}^\dagger _\xi {f}\) is well defined \({\mathfrak {w}}\)-almost everywhere and
The \(\Vert \cdot \Vert _\chi \)-norm of integrant in (45) is bounded by \(\exp \left\{ \varepsilon \Vert \xi \Vert _{w_0}^2\right\} \) with any \(\varepsilon >0\). By Fernique’s theorem and (48), the integral (45) with the Wiener measure \({\mathfrak {w}}\) exists for all \(f\in {L}^{+2}_\chi \). The equality \({\mathfrak {w}}(w_0)=1\) implies that the integral (45) is absolutely convergent uniformly in variables \(r>0\) on the whole space \(c_0\). It provides the \(C_0\)-property of \({\mathfrak {G}}_r\) in variables \(r>0\) on any finite sum \(\bigoplus _{m=0}^nL^{2,m}_\chi \).
Prove that the semigroup \({\mathfrak {G}}_r\) is generated by \(\sum {\mathfrak {p}}_m^{\dagger 2}\) with \({\mathfrak {p}}^\dagger _m:={\mathbb {i}(\partial _m^\dagger +{{\bar{\phi }}}_m)}\). By differentiation of \({W}^\dagger (\mathbb {i}\xi _m{a},\mathbb {i}\xi _m{a})\) at \({\xi _m=0}\), we get that its generator coincides with \({\mathfrak {p}}^\dagger _m\). In fact, \({W}^\dagger (\mathbb {i}\xi _m{a},\mathbb {i}\xi _m{a}){f}=\exp \big \{\xi _m{\mathfrak {p}}_m^\dagger \big \}f\) for all \({f\in \phi ^{\mathbb {Y}}}\). Applying the next formula for Gamma functions with \(\alpha =(\alpha _1,\ldots ,\alpha _n)\in {\mathbb {N}}_0^n\)
we find that for any \({L}^{+2}_\chi \)-valued cylinder function \(h_n=({W}^\dagger _\xi {f})\circ p_n^\sim \) we have
Using (48), we obtain that \(0\le r\longmapsto {\mathfrak {G}}_r^\dagger \) is the 1-parameter \(C_0\)-semigroup on any finite sum \(\bigoplus _{m=0}^nL^{2,m}_\chi \) with densely defined closed generator \(\sum _{m=1}^{n}{\mathfrak {p}}_m^{\dagger 2}\). Applying the known relation [32] between the initial problem (46) and the 1-parameter \(C_0\)-semigroup \({\mathfrak {G}}_r^\dagger \), we obtain that the function \(w_n(r)={\mathfrak {G}}^\dagger _rf_n\) for any \({n\in {\mathbb {N}}}\) solves this problem in the sense that \(d{\mathfrak {G}}^\dagger _rf_n/dr|_{r=0}=\sum _{m=1}^{n}{\mathfrak {p}}_m^{\dagger 2}f_n\) for all \({f_n\in }\bigoplus _{m=0}^nL^{2,m}_\chi \). The theorem is proved. \(\square \)
Taking into account the isometries \({H}^2_\beta {\mathop {\simeq }\limits ^{\varPsi }}{L}^2_\chi \) and \({P}_\beta ^n(H){\mathop {\simeq }\limits ^{\varPsi }}{L}^{2,n}_\chi \) from (28), defined by linearization, we can rewrite the Cauchy problem in polynomial form.
Consider the Weyl system \({W(a,b)=\exp \left\{ \langle {a}\mid {b}\rangle /2 \right\} M_{b^*}T_a}\) defined by (43) on the dense subspace of polynomials \({P}_\beta (H):={\sum }_{n\ge 0}{P}_\beta ^n(H)\) in \({H}^2_\beta ,\) consisting of all finite sums of n-homogenous polynomials \(\psi ^*(h)=\sum \psi _n^*(h)\) of variable \({h\in H}\) with components \(\psi _n^*={\mathcal {P}}\circ \psi _n\in {P}_\beta ^n(H)\). Replacing a by \(\tau a\) and b by \(\tau b\) with real \({\tau \in {\mathbb {R}}}\), we get that \(T_{\tau a}\) and \(M_{\tau b^*}\) are generated by closed generators on \({P}_\beta (H)\),
As a consequence, the 1-parameter Weyl system \({W}(\tau {a},\tau {b})\) has the generator
densely defined on \({P}_\beta (H)\) such that \((\tau b)^*+\partial _{\tau a}^*=\tau (b^*+\partial _a^*)\) for real \(\tau \). Let \({W}_{p^\sim _n(\xi )}= \prod _{m=1}^{n}{W}(\mathbb {i}\xi _m{\mathfrak {e}}_m,\mathbb {i}\xi _m{\mathfrak {e}}_m)\), \(T_{p^\sim _n(\xi )}=\prod _{m=1}^{n} T_{\mathbb {i}\xi _m{\mathfrak {e}}_m}\), \(M_{p^\sim _n(\xi )}=\prod _{m=1}^{n} M_{-\mathbb {i}\xi _m{\mathfrak {e}}_m^*}\).
Corollary 6
For all \(\psi ^*\in {P}_\beta (H)\) and \(\xi =(\xi _m)\in c_0\) there exists the limit
\({\mathfrak {w}}\)-almost everywhere on \(c_0\) such that the 1-parameter Gaussian semigroup
is generated by \(-\sum ({\mathfrak {e}}_m^*+\partial _m^*)^2\). Thus, \(w(r)={\mathfrak {G}}_r\psi ^*\) is unique solution of the problem
in the space of Hilbert–Schmidt polynomials \({P}_\beta (H)\).
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Lopushansky, O. Weyl–Schrödinger Representations of Heisenberg Groups in Infinite Dimensions. Results Math 75, 73 (2020). https://doi.org/10.1007/s00025-020-01198-0
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DOI: https://doi.org/10.1007/s00025-020-01198-0
Keywords
- Infinite-dimensional Heisenberg group
- Weyl–Schrödinger representation in infinite dimensions
- Schur polynomials on Paley–Wiener maps
- Fourier analysis on virtual unitary matrices
- heat equation over Heisenberg group