scholarly journals Toeplitz Operators with Horizontal Symbols Acting on the Poly-Fock Spaces

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Armando Sánchez-Nungaray ◽  
Carlos González-Flores ◽  
Raquiel Rufino López-Martínez ◽  
Jorge Luis Arroyo-Neri
Keyword(s):  
Complex Plane ◽  
Gaussian Measure ◽  
Toeplitz Operators ◽  
Fock Space ◽  
Identity Matrix ◽  
Fock Spaces ◽  
Limit Values ◽  
Bounded Functions

We describe the C⁎-algebra generated by the Toeplitz operators acting on each poly-Fock space of the complex plane C with the Gaussian measure, where the symbols are bounded functions depending only on x=Re  z and have limit values at y=-∞ and y=∞. The C⁎ algebra generated with this kind of symbols is isomorphic to the C⁎-algebra functions on extended reals with values on the matrices of dimension n×n, and the limits at y=-∞ and y=∞ are scalar multiples of the identity matrix.

scholarly journals Toeplitz Operators with Lagrangian Invariant Symbols Acting on the Poly-Fock Space of ℂ n

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Jorge Luis Arroyo Neri ◽  
Armando Sánchez-Nungaray ◽  
Mauricio Hernández Marroquin ◽  
Raquiel R. López-Martínez
Keyword(s):  
Toeplitz Operators ◽  
Complex Space ◽  
Fock Space ◽  
Identity Matrix ◽  
The Matrix

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.

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

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.

scholarly journals Holomorphic Fock spaces for positive linear transformations

2006 ◽  
Vol 98 (2) ◽  
pp. 262 ◽  
Author(s):  
R. Fabec ◽  
G. Ólafsson ◽  
A.N. Sengupta
Keyword(s):  
Gaussian Measure ◽  
Reproducing Kernel ◽  
Fock Space ◽  
Inner Product ◽  
Fock Spaces ◽  
Linear Transformations ◽  
Positive Real ◽  
Bargmann Transform ◽  
Finite Dimensional ◽  
Real Linear

Suppose $A$ is a positive real linear transformation on a finite dimensional complex inner product space $V$. The reproducing kernel for the Fock space of square integrable holomorphic functions on $V$ relative to the Gaussian measure $d\mu_A(z)=\frac {\sqrt{\det A}} {\pi^n}e^{-\Re\langle Az,z\rangle}\,dz$ is described in terms of the linear and antilinear decomposition of the linear operator $A$. Moreover, if $A$ commutes with a conjugation on $V$, then a restriction mapping to the real vectors in $V$ is polarized to obtain a Segal-Bargmann transform, which we also study in the Gaussian-measure setting.

scholarly journals CUMULANTS IN NONCOMMUTATIVE PROBABILITY THEORY III: CREATION AND ANNIHILATION OPERATORS ON FOCK SPACES

2005 ◽  
Vol 08 (03) ◽  
pp. 407-437 ◽  
Author(s):  
FRANZ LEHNER
Keyword(s):  
Probability Theory ◽  
Symmetric Group ◽  
Toeplitz Operators ◽  
Fock Space ◽  
Random Variables ◽  
Fock Spaces ◽  
Noncommutative Probability ◽  
Infinite Symmetric Group ◽  
Fock States ◽  
Cycle Indicator

Exchangeability systems arising from Fock space constructions are considered and the corresponding cumulants are computed for generalized Toeplitz operators and similar noncommutative random variables. In particular, simplified calculations are given for the two known examples of q-cumulants. In the second half of the paper we consider in detail the Fock states associated to characters of the infinite symmetric group recently constructed by Bożejko and Guta. We express moments of multidimensional Dyck words in terms of the so-called cycle indicator polynomials of certain digraphs.

scholarly journals Toeplitz operators between large Fock spaces in several complex variables

2021 ◽  
Vol 7 (1) ◽  
pp. 1293-1306
Author(s):  
Ermin Wang ◽  
◽  
Jiajia Xu
Keyword(s):  
Toeplitz Operator ◽  
Toeplitz Operators ◽  
Borel Measure ◽  
Fock Space ◽  
Holomorphic Functions ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Weight Class ◽  
Fock Spaces ◽  
Positive Borel Measure

<abstract><p>Let $ \omega $ belong to the weight class $ \mathcal{W} $, the large Fock space $ \mathcal{F}_{\omega}^{p} $ consists of all holomorphic functions $ f $ on $ \mathbb{C}^{n} $ such that the function $ f(\cdot)\omega(\cdot)^{1/2} $ is in $ L^p(\mathbb{C}^{n}, dv) $. In this paper, given a positive Borel measure $ \mu $ on $ {\mathbb C}^n $, we characterize the boundedness and compactness of Toeplitz operator $ T_\mu $ between two large Fock spaces $ F^{p}_\omega $ and $ F^{q}_\omega $ for all possible $ 0 &lt; p, q &lt; \infty $.</p></abstract>

Toeplitz Operators on the Fock Space with Presymbols Discontinuous on a Thick Set

1996 ◽  
Vol 180 (1) ◽  
pp. 299-315 ◽  
Author(s):  
E. Ram Írez De Arellano ◽  
N. L. Vasilevski
Keyword(s):  
Toeplitz Operators ◽  
Fock Space

scholarly journals Factorization of the canonical bases for higher-level Fock spaces

2011 ◽  
Vol 55 (1) ◽  
pp. 23-51 ◽  
Author(s):  
Susumu Ariki ◽  
Nicolas Jacon ◽  
Cédric Lecouvey
Keyword(s):  
Fock Space ◽  
Fock Spaces ◽  
Transition Matrices ◽  
Canonical Bases ◽  
Highest Weight ◽  
Highest Weight Modules ◽  
Graphic Module ◽  
Module Structures ◽  
Weight Modules

AbstractThe level l Fock space admits canonical bases $\mathcal{G}_{e}$ and $\smash{\mathcal{G}_{\infty}}$. They correspond to $\smash{\mathcal{U}_{v}(\widehat{\mathfrak{sl}}_{e})}$ and $\mathcal{U}_{v}({\mathfrak{sl}}_{\infty})$-module structures. We establish that the transition matrices relating these two bases are unitriangular with coefficients in ℕ[v]. Restriction to the highest-weight modules generated by the empty l-partition then gives a natural quantization of a theorem by Geck and Rouquier on the factorization of decomposition matrices which are associated to Ariki–Koike algebras.

scholarly journals Interacting Fock spaces and subproduct systems

2020 ◽  
Vol 23 (03) ◽  
pp. 2050017
Author(s):  
Malte Gerhold ◽  
Michael Skeide
Keyword(s):  
Selfadjoint Operator ◽  
Operator Algebras ◽  
Fock Space ◽  
Fock Spaces ◽  
Full Generality ◽  
Necessary And Sufficient Criteria ◽  
Definition Of ◽  
Interacting Fock Space ◽  
Necessary And Sufficient

We present a new more flexible definition of interacting Fock space that allows to resolve in full generality the problem of embeddability. We show that the same is not possible for regularity. We apply embeddability to classify interacting Fock spaces by squeezings. We give necessary and sufficient criteria for when an interacting Fock space has only bounded creators, giving thus rise to new classes of non-selfadjoint and selfadjoint operator algebras.

scholarly journals Hyponormal Toeplitz operators on H2(T) with polynomial symbols

1996 ◽  
Vol 144 ◽  
pp. 179-182 ◽  
Author(s):  
Dahai Yu
Keyword(s):  
Functional Equation ◽  
Hardy Space ◽  
Closed Form ◽  
Unit Circle ◽  
Toeplitz Operator ◽  
Explicit Expression ◽  
Complex Plane ◽  
Toeplitz Operators ◽  
Computing Process

Let T be the unit circle on the complex plane, H2(T) be the usual Hardy space on T, Tø be the Toeplitz operator with symbol Cowen showed that if f1 and f2 are functions in H such that is in Lø, then Tf is hyponormal if and only if for some constant c and some function g in H∞ with Using it, T. Nakazi and K. Takahashi showed that the symbol of hyponormal Toeplitz operator Tø satisfies and and they described the ø solving the functional equation above. Both of their conditions are hard to check, T. Nakazi and K. Takahashi remarked that even “the question about polynomials is still open” [2]. Kehe Zhu gave a computing process by way of Schur’s functions so that we can determine any given polynomial ø such that Tø is hyponormal [3]. Since no closed-form for the general Schur’s function is known, it is still valuable to find an explicit expression for the condition of a polynomial á such that Tø is hyponormal and depends only on the coefficients of ø, here we have one, it is elementary and relatively easy to check. We begin with the most general case and the following Lemma is essential.

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