Fock Space, the Heisenberg Group, Heat Flow, and Toeplitz Operators

We introduce the so-called extended Lagrangian symbols, and we prove that the C ∗ -algebra generated by Toeplitz operators with these kind of symbols acting on the homogeneously poly-Fock space of the complex space ℂ n is isomorphic and isometric to the C ∗ -algebra of matrix-valued functions on a certain compactification of ℝ n obtained by adding a sphere at the infinity; moreover, the matrix values at the infinity points are equal to some scalar multiples of the identity matrix.

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Toeplitz Operators with I M O s Symbols between Generalized Fock Spaces

Journal of Function Spaces ◽

10.1155/2021/8044854 ◽

2021 ◽

Vol 2021 ◽

pp. 1-9

Author(s):

Yiyuan Zhang ◽

Guangfu Cao ◽

Li He

Keyword(s):

Toeplitz Operators ◽

Fock Space ◽

Fock Spaces

In this paper, we study the mapping properties of Toeplitz operators T f associated with IMO s symbols f acting between two generalized Fock spaces F φ p , where 1 < s ≤ ∞ . We characterize bounded or compact Toeplitz operators T f from one generalized Fock space F φ p to another F φ q , respectively, in four cases.

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Toeplitz operators on generalized Bergman-Hardy spaces

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-14316 ◽

2001 ◽

Vol 88 (1) ◽

pp. 96

Author(s):

Wolfgang Lusky

Keyword(s):

Hardy Spaces ◽

Toeplitz Operators ◽

Fock Space ◽

Bergman Spaces ◽

Trigonometric Polynomials ◽

Essential Spectra ◽

Radial Functions ◽

Hardy And Bergman Spaces ◽

Schmidt Operator ◽

Vector Valued

We study the Toeplitz operators $T_f: H_2 \to H_2$, for $f \in L_\infty$, on a class of spaces $H_2$ which in- cludes, among many other examples, the Hardy and Bergman spaces as well as the Fock space. We investigate the space $X$ of those elements $f \in L_\infty$ with $\lim_j \|T_f-T_{f_j}\|=0$ where $(f_j)$ is a sequence of vector-valued trigonometric polynomials whose coefficients are radial functions. For these $T_f$ we obtain explicit descriptions of their essential spectra. Moreover, we show that $f \in X$, whenever $T_f$ is compact, and characterize these functions in a simple and straightforward way. Finally, we determine those $f \in L_\infty$ where $T_f$ is a Hilbert-Schmidt operator.

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The Vacuum Distributions of the Truncated Virasoro Fields are Products of Gamma Distributions

Open Systems & Information Dynamics ◽

10.1142/s1230161217500044 ◽

2017 ◽

Vol 24 (01) ◽

pp. 1750004 ◽

Cited By ~ 1

Author(s):

Luigi Accardi ◽

Andreas Boukas ◽

Yun-Gang Lu

Keyword(s):

Fourier Transform ◽

Characteristic Function ◽

Heisenberg Group ◽

Lie Algebra ◽

Fock Space ◽

Random Variables ◽

Virasoro Algebra ◽

Commutation Relations ◽

Gamma Distributions

In a recent paper, using a splitting formula for the multi-dimensional Heisenberg group, we derived a formula for the vacuum characteristic function (Fourier transform) of quantum random variables defined as self-adjoint sums of Fock space operators satisfying the multidimensional Heisenberg Lie algebra commutation relations. In this paper we use that formula to compute the characteristic function of quantum random variables defined as suitably truncated sums of the Virasoro algebra generators. By relating the structure of the Virasoro fields to the quadratic quantization program and using techniques developed in that context we prove that the vacuum distributions of the truncated Virasoro fields are products of independent, but not identically distributed, shifted Gamma-random variables.

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