Perelomov problem and inversion of the Segal-Bargmann transform

Segal-Bargmann Transform and Paley-Wiener Theorems on Motion Groups

We study the Segal-Bargmann transform on a motion group Rn n K; where K is a compact subgroup of SO(n): A characterization of the Poisson integrals associated to the Laplacian on Rn n K is given. We also establish a Paley-Wiener type theorem using the complexified representations.

Segal-Bargmann transform and holomorphic Sobolev spaces of fractional order

In this note we investigate the image of Sobolev spaces of fractional order on a compact Lie group $ K $ under the Segal-Bargmann transform. We show that the image can be characterised in terms of certain weighted Bergman spaces of holomorphic functions on the complexification extending a theorem of Hall and Lewkeeratiyutkul. We also treat the heat kernel transform associated to the Hermite operator.

Segal-Bargmann transform and Paley-Wiener theorems on $M(2).$

We study the Segal-Bargmann transform on $M(2).$ The range of this transform is characterized as a weighted Bergman space. In a similar fashion Poisson integrals are studied. Using a Gutzmer type formula we characterize the range as a class of functions extending holomorphically to an appropriate domain in the complexification of $M(2).$ We also prove a Paley-Wiener theorem for the inverse Fourier transform

Bounded Berezin-Toeplitz operators on the Segal-Bargmann space

We discuss the boundedness of Berezin-Toeplitz operators on a generalized Segal-Bargmann space (Fock space) over the complex $n$-space. This space is characterized by the image of a global Bargmann-type transform introduced by Sjöstrand. We also obtain the deformation estimates of the composition of Berezin-Toeplitz operators whose symbols and their derivatives up to order three are in the Wiener algebra of Sjöstrand. Our method of proofs is based on the pseudodifferential calculus and the heat flow determined by the phase function of the Bargmann transform.

Segal-Bargmann transform and Paley-Wiener theorems on Heisenberg motion groups

We study the Segal-Bargmann transform on the Heisenberg motion groups $\mathbb{H}^n \ltimes K,$ where $\mathbb{H}^n$ is the Heisenberg group and $K$ is a compact subgroup of $U(n)$ such that $(K,\mathbb{H}^n)$ is a Gelfand pair. The Poisson integrals associated to the Laplacian for the Heisenberg motion group are also characterized using Gutzmer's formulae. Explicitly realizing certain unitary irreducible representations of $\mathbb{H}^n \ltimes K,$ we prove the Plancherel theorem. A Paley-Wiener type theorem is proved using complexified representations.

Duality in Segal-Bargmann Spaces

For $α>0$, the Bargmann projection $P_α$ is the orthogonal projection from $L^2(γ_α)$ onto the holomorphic subspace $L^2_{hol}(γ_α)$, where $γ_α$ is the standard Gaussian probability measure on $\C^n$ with variance $(2α)^{-n}$. The space $L^2_{hol}(γ_α)$ is classically known as the Segal-Bargmann space. We show that $P_α$ extends to a bounded operator on $L^p(γ_{αp/2})$, and calculate the exact norm of this scaled $L^p$ Bargmann projection. We use this to show that the dual space of the $L^p$-Segal-Bargmann space $L^p_{hol}(γ_{αp/2})$ is an $L^{p'}$ Segal-Bargmann space, but with the Gaussian …

Clifford Algebra-valued Segal-Bargmann Transform and Taylor Isomorphism

Classical Segal-Bargmann theory studies three Hilbert space unitary isomorphisms that describe the wave-particle duality and the configuration space-phase space. In this work, we generalized these concepts to Clifford algebra-valued functions. We establish the unitary isomorphisms among the space of Clifford algebra-valued square-integrable functions on $\mathbb{R}^n$ with respect to a Gaussian measure, the space of monogenic square-integrable functions on $\mathbb{R}^{n+1}$ with respect to another Gaussian measure and the space of Clifford algebra-valued linear functionals on symmetric tensor…

Schatten class Hankel operators on the Segal-Bargmann space and the Berger-Coburn phenomenon

The authors provide a correct proof of Theorem 1.2 and correct the statement of Theorem 2.6 in their paper, which appeared in Trans. Amer. Math. Soc. 375 (2022), 3733-3753.

Segal-Bargmann 空间上范数可达的复合算子

Norm attaining composition operators on Segal-Bargmann spaces

本文研究 Segal-Bargmann 空间上复合算子的范数可达性，证明 $\mathbb{C}^n$ 上 Fock 空间中所有复合算子均范数可达，并给出两个核函数之和为极值函数的充要条件。

Segal-Bargmann transforms from hyperbolic Hamiltonians

We consider the imaginary time flow of a quadratic hyperbolic Hamiltonian on the symplectic plane, apply it to the Schrödinger polarization and study the corresponding evolution of polarized sections. The flow is periodic in imaginary time and the evolution of polarized sections has interesting features. On the time intervals for which the polarization is real or Kähler, the half--form corrected time evolution of polarized sections is given by unitary operators which turn out to be equivalent to the classical Segal-Bargmann transforms (which are usually associated to the quadratic elliptic Ham…

Boundedness and compactness of composition operators on Segal-Bargmann spaces

For $E$ a Hilbert space, let $\mathcal{H}(E)$ denote the Segal-Bargmann space (also known as the Fock space) over $E$, which is a reproducing kernel Hilbert space with kernel $K(x,y)=\exp(< x,y>)$ for $x,y$ in $E$. If $ϕ$ is a mapping on $E$, the composition operator $C_ϕ$ is defined by $C_ϕh = h\circϕ$ for $h\in \mathcal{H}(E)$ for which $h\circϕ$ also belongs to $\mathcal{H}(E)$. We determine necessary and sufficient conditions for the boundedness and compactness of $C_ϕ$. Our results generalize results obtained earlier by Carswell, MacCluer and Schuster for finite dimensional spaces $E$.

Holomorphic Hermite functions in Segal-Bargmann spaces

We study systems of holomorphic Hermite functions in the Segal-Bargmann spaces, which are Hilbert spaces of entire functions on the complex Euclidean space, and are determined by the Bargmann-type integral transform on the real Euclidean space. We prove that for any positive parameter which is strictly smaller than the minimum eigenvalue of the positive Hermitian matrix associated with the transform, one can find a generator of holomorphic Hermite functions whose anihilation and creation operators satisfy canonical commutation relations. In other words, we find the necessary and sufficient con…

The Segal-Bargmann Transform on Classical Matrix Lie Groups

We study the complex-time Segal-Bargmann transform $\mathbf{B}_{s,τ}^{K_N}$ on a compact type Lie group $K_N$, where $K_N$ is one of the following classical matrix Lie groups: the special orthogonal group $\mathrm{SO}(N,\mathbb{R})$, the special unitary group $\mathrm{SU}(N)$, or the compact symplectic group $\mathrm{Sp}(N)$. Our work complements and extends the results of Driver, Hall, and Kemp on the Segal-Bargman transform for the unitary group $\mathrm{U}(N)$. We provide an effective method of computing the action of the Segal-Bargmann transform on \emph{trace polynomials}, which comprise …

Cyclic Composition Operators on Segal-Bargmann space

We study the hypercyclic, supercyclic and cyclic properties of composition operator $C_ϕ$ on the Segal-Bargmann space $\mathscr{H}(\mathscr{E})$, where $ϕ(z)=Az+b$, $A\in \mathcal{B}(\mathscr{E})$, $b\in \mathscr{E}$ with $\left\|A\right\|\leq 1$ and $A^*b\in (I-A^*A)^{\frac{1}{2}}$. In this connection we also give a characterization of the symbols $ϕ$ which induce the bounded composition operator $C_ϕ$ on $\mathscr{H}(\mathscr{E})$ and show that the properties of $ϕ$ influence the cyclic behaviour of $C_ϕ$.

The Large-$N$ Limit of the Segal--Bargmann Transform on $\mathbb{U}_N$

We study the (two-parameter) Segal--Bargmann transform $\mathbf{B}_{s,t}^N$ on the unitary group $\mathbb{U}_N$, for large $N$. Acting on matrix valued functions that are equivariant under the adjoint action of the group, the transform has a meaningful limit $\mathscr{G}_{s,t}$ as $N\to\infty$, which can be identified as an operator on the space of complex Laurent polynomials. We introduce the space of {\em trace polynomials}, and use it to give effective computational methods to determine the action of the heat operator, and thus the Segal--Bargmann transform. We prove several concentration o…

Large-$N$ Limit of the Segal--Bargmann Transforms on the Spheres

We study the large-$N$ limit of the Segal--Bargmann transform on $S^{N-1}(\sqrt N)$, the $(N-1)$-dimensional sphere of radius $\sqrt N$, as a unitary map from the space of square-integrable functions with respect to the normalized spherical measure onto the space of holomorphic square-integrable functions with respect to a certain measure on the quadric. In particular, we give an explicit formulation and describe the geometric models for the limit of the domain, the limit of the range, and the limit of the transform when $N$ tends to infinity. We show that the limiting transform is still a uni…

On Sharp Constants for Dual Segal--Bargmann $L^p$ Spaces

We study dilated holomorphic $L^p$ space of Gaussian measures over $\mathbb{C}^n$, denoted $\mathcal{H}_{p,α}^n$ with variance scaling parameter $α>0$. The duality relations $(\mathcal{H}_{p,α}^n)^\ast \cong \mathcal{H}_{p',α}$ hold with $\frac{1}{p}+\frac{1}{p'}=1$, but not isometrically. We identify the sharp lower constant comparing the norms on $\mathcal{H}_{p',α}$ and $(\mathcal{H}_{p,α}^n)^\ast$, and provide upper and lower bounds on the sharp upper constant. We prove several suggestive partial results on the sharpness of the upper constant. One of these partial results leads to a sharp …

通过巴尔曼-塞格尔范数刻画（分数）马利文-瓦坦贝-索波列夫空间 $\mathcal{D}^{α,2}$

Characterization of the (fractional) Malliavin-Watanabe-Sobolev spaces $\mathcal{D}^{α,2}$ via the Bargmann-Segal norm

本文通过巴尔曼-塞格尔范数刻画了分数阶马利文-瓦坦贝-索波列夫空间$\mathcal{D}^{α,2}$，并给出了正则性和负则性的判据，连接了马利文微积分与白噪声分析技术。

Segal-Bargmann transform on Hermitian symmetric spaces and Orthogonal Polynomials

Let $\mathcal{D}=G/K$ be a complex bounded symmetric domain of tube type in a complex Jordan algebra $V$ and let $\mathcal{D}_{\mathbb{R}}=H/L\subset \mathcal{D}$ be its real form in a formally real Euclidean Jordan algebra $J\subset V$. We consider representations of $H$ that are gotten by the generalized Segal-Bargmann transform from a unitary $G$-space of holomorphic functions on $\mathcal{D}$ to an $L^2$-space on $\mathcal{D_{\mathbf{R}}}$. We prove that in the unbounded realization the inverse of the unitary part of the restriction map is actually the Laplace transform. We find the extens…

Cross-Toeplitz Operators on the Fock--Segal--Bargmann Spaces and Two-Sided Convolutions on the Heisenberg Group

We introduce an extended class of cross-Toeplitz operators which act between Fock--Segal--Bargmann spaces with different weights. It is natural to consider these operators in the framework of representation theory of the Heisenberg group. Our main technique is representation of cross-Toeplitz by two-sided relative convolutions from the Heisenberg group. In turn, two-sided convolutions are reduced to usual (one-sided) convolutions on the Heisenberg group of the doubled dimensionality. This allows us to utilise the powerful group-representation technique of coherent states, co- and contra-varian…

利用Touchard多项式与超几何函数研究Bargmann变换的新特性

New aspects of Bargmann transform using Touchard polynomials and hypergeometric functions

本文研究Schwartz空间及其对偶在Segal-Bargmann变换下的像空间，刻画其对应的再生核Hilbert空间，引入两种广义Bargmann变换，延续了Neretin关于高斯积分算子的研究路径。

The Complex-Time Segal-Bargmann Transform

We introduce a new form of the Segal--Bargmann transform for a Lie group $K$ of compact type. We show that the heat kernel $(ρ_{t}(x))_{t>0,x\in K}$ has a space-time analytic continuation to a holomorphic function \[ (ρ_{\mathbb{C}}(τ,z))_{\mathrm{Re}\,τ>0,z\in K_{\mathbb{C}}} \] where $K_{\mathbb{C}}$ is the complexification of $K$. The new transform is defined by the integral \[ (B_τf)(z)=\int_{K}ρ_{\mathbb{C}}(τ,zk^{-1})f(k)\,dk,\quad z\in K_{\mathbb{C}}. \] If $s>0$ and $τ\in\mathbb{D}(s,s)$ (the disk of radius $s$ centered at $s$), this integral defines a holomorphic function on $K_{\math…

Radial operators on polyanalytic Bargmann-Segal-Fock spaces

The paper considers bounded linear radial operators on the polyanalytic Fock spaces $\mathcal{F}_n$ and on the true-polyanalytic Fock spaces $\mathcal{F}_{(n)}$. The orthonormal basis of normalized complex Hermite polynomials plays a crucial role in this study; it can be obtained by the orthogonalization of monomials in $z$ and $\overline{z}$. First, using this basis, we decompose the von Neumann algebra of radial operators, acting in $\mathcal{F}_n$, into the direct sum of some matrix algebras, i.e. radial operators are represented as matrix sequences. Secondly, we prove that the radial opera…

Wavelets in Banach Spaces

We describe a construction of wavelets (coherent states) in Banach spaces generated by ``admissible'' group representations. Our main targets are applications in pure mathematics while connections with quantum mechanics are mentioned. As an example we consider operator valued Segal-Bargmann type spaces and the Weyl functional calculus. Keywords: Wavelets, coherent states, Banach spaces, group representations, covariant, contravariant (Wick) symbols, Heisenberg group, Segal-Bargmann spaces, Weyl functional calculus (quantization), second quantization, bosonic field.

Generalized Fock spaces and the Stirling numbers

The Bargmann-Fock-Segal space plays an important role in mathematical physics, and has been extended into a number of directions. In the present paper we imbed this space into a Gelfand triple. The spaces forming the Fréchet part (i.e. the space of test functions) of the triple are characterized both in a geometric way and in terms of the adjoint of multiplication by the complex variable, using the Stirling numbers of the second kind. The dual of the space of test functions has a topological algebra structure, of the kind introduced and studied by the first named author and G. Salomon.

Toeplitz Operators in a Symmetrically-Normed Ideal

We look at Toeplitz operators $T_ν$ on the Fock Space (also known as the Segal-Bargmann space) which have a positive Borel measure $ν$ as a symbol. We characterize when $\left(T_ν\right)^s$ for $0<s\leq 1$ is in the symmetrically normed ideal associated with any given symmetric norming function.

Coherent State Transforms and the Weyl Equation in Clifford Analysis

We study a transform, inspired by coherent state transforms, from the Hilbert space of Clifford algebra valued square integrable functions $L^2({\mathbb R}^m,dx)\otimes {\mathbb C}_{m}$ to a Hilbert space of solutions of the Weyl equation on ${\mathbb R}^{m+1}= {\mathbb R} \times {\mathbb R}^m$, namely to the Hilbert space ${\mathcal M}L^2({\mathbb R}^{m+1},dμ)$ of ${\mathbb C}_m$-valued monogenic functions on ${\mathbb R}^{m+1}$ which are $L^2$ with respect to an appropriate measure $dμ$. We prove that this transform is a unitary isomorphism of Hilbert spaces and that it is therefore an analo…