scholarly journals Distributions for Nonsymmetric Monotone and Weakly Monotone Position Operators

2021 ◽  
Vol 15 (6) ◽  
Author(s):  
Vitonofrio Crismale ◽  
Maria Elena Griseta ◽  
Janusz Wysoczański
Keyword(s):  
Fock Space ◽  
Recursive Formula ◽  
Partial Sums ◽  
Fock Spaces ◽  
Continuous Part ◽  
Absolutely Continuous ◽  
Single Operator ◽  
Meixner Class ◽  
Finite Sum ◽  
Poisson Type

AbstractWe study the vacuum distribution, under an appropriate scaling, of a family of partial sums of nonsymmetric position operators on weakly monotone and monotone Fock spaces, respectively. We preliminary treat the case of weakly monotone Fock space, and show that any single operator has the vacuum law belonging to the free Meixner class. After establishing some relations between the combinatorics of Motzkin and Riordan paths, we give a recursive formula for the vacuum moments of the law of any finite sum. Since the operators are monotone independent, the distribution is the monotone convolution of the free Meixner law above. We also investigate the asymptotic measure for these sums, which can be seen as “Poisson type” limit law. It turns out to belong to the free Meixner class, with an atomic and an absolutely continuous part (w.r.t. the Lebesgue measure). Finally, we briefly apply analogous considerations to the case of monotone Fock space.

A d-DEFORMATION OF FREE GAUSSIAN RANDOM VARIABLES

2005 ◽  
Vol 08 (04) ◽  
pp. 669-680 ◽  
Author(s):  
JANUSZ WYSOCZAŃSKI
Keyword(s):  
Tensor Product ◽  
Hypergeometric Series ◽  
Fock Space ◽  
Recurrence Formula ◽  
Random Variables ◽  
Product Formula ◽  
Continuous Part ◽  
Absolutely Continuous ◽  
Gaussian Random Variables ◽  
Special Case

We define a deformation of free creations (and annihilations), given by operators on the full Fock space, acting nontrivially only between the vacuum subspace ℂΩ and the twofold tensor product [Formula: see text]. Then we study the distribution of the deformed free gaussian operators, with the deformation containing also a real parameter d. The recurrence formula for moments is shown, and the Cauchy transform of the distribution measure is computed. This yields the description of the measure: absolutely continuous part and the atomic part. The existence of atoms depends on the parameter d. The special case d =1 is studied with all details, with the formula for moments is given as values of the hypergeometric series. Finally we show the formula for computing the mixed moments of the deformed operators.

MONOTONIC INDEPENDENCE ON THE WEAKLY MONOTONE FOCK SPACE AND RELATED POISSON TYPE THEOREM

2005 ◽  
Vol 08 (02) ◽  
pp. 259-275 ◽  
Author(s):  
JANUSZ WYSOCZAŃSKI
Keyword(s):  
Limit Theorem ◽  
Fock Space ◽  
Type Theorem ◽  
Random Variables ◽  
Direct Proof ◽  
Binomial Coefficients ◽  
Second Quantization ◽  
The Central Limit Theorem ◽  
Natural Way ◽  
Poisson Type

We show how the construction of t-transformation can be applied to the construction of a sequence of monotonically independent noncommutative random variables. We introduce the weakly monotone Fock space, on which these operators act. This space can be derived in a natural way from the papers by Pusz and Woronowicz on twisted second quantization. It was observed by Bożejko that, by taking μ = 0, for the μ-CAR relations one obtains the Muraki's monotone Fock space, while for the μ-CCR relations one obtains the weakly monotone Fock space. We show that the direct proof of the central limit theorem for these operators provides an interesting recurrence for the highest binomial coefficients. Moreover, we show the Poisson type theorem for these noncommutative random variables.

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 Deformed Fock spaces, Hecke operators and monotone Fock space of Muraki

2012 ◽  
Vol 45 (2) ◽  
Author(s):  
Marek Bożejko
Keyword(s):  
Fock Space ◽  
Hecke Operators ◽  
Fock Spaces

AbstractThe main purpose of this paper is to extend our previous construction of

scholarly journals PARAFERMIONS, PARABOSONS AND REPRESENTATIONS OF 𝔰𝔬(∞) AND 𝔬𝔰𝔭(1|∞)

2009 ◽  
Vol 20 (06) ◽  
pp. 693-715 ◽  
Author(s):  
N. I. STOILOVA ◽  
J. VAN DER JEUGT
Keyword(s):  
Lie Algebra ◽  
Fock Space ◽  
Complete Solution ◽  
Lie Superalgebra ◽  
Explicit Construction ◽  
Fock Spaces ◽  
Infinite Rank ◽  
Infinite Set ◽  
Basis Vectors

The goal of this paper is to give an explicit construction of the Fock spaces of the parafermion and the paraboson algebra, for an infinite set of generators. This is equivalent to constructing certain unitary irreducible lowest weight representations of the (infinite rank) Lie algebra 𝔰𝔬(∞) and of the Lie superalgebra 𝔬𝔰𝔭(1|∞). A complete solution to the problem is presented, in which the Fock spaces have basis vectors labelled by certain infinite but stable Gelfand–Zetlin patterns, and the transformation of the basis is given explicitly. We also present expressions for the character of the Fock space representations.

scholarly journals Wavelet transforms in generalized Fock spaces

1997 ◽  
Vol 20 (4) ◽  
pp. 657-672 ◽  
Author(s):  
John Schmeelk ◽  
Arpad Takaci
Keyword(s):  
Wavelet Transform ◽  
Wavelet Transforms ◽  
Fock Space ◽  
Generalized Function ◽  
Fock Spaces ◽  
The Individual ◽  
Individual Entry

A generalized Fock space is introduced as it was developed by Schmeelk [1-5], also Schmeelk and Takači [6-8]. The wavelet transform is then extended to this generalized Fock space. Since each component of a generalized Fock functional is a generalized function, the wavelet transform acts upon the individual entry much the same as was developed by Mikusinski and Mort [9] based upon earlier work of Mikusinski and Taylor [10]. It is then shown that the generalized wavelet transform applied to a member of our generalized Fock space produces a more appropriate functional for certain appfications.

scholarly journals LOCAL EXPONENTS AND INFINITESIMAL GENERATORS OF CANONICAL TRANSFORMATIONS ON BOSON FOCK SPACES

2004 ◽  
Vol 07 (04) ◽  
pp. 547-571 ◽  
Author(s):  
F. HIROSHIMA ◽  
K. R. ITO
Keyword(s):  
Canonical Transformation ◽  
Symplectic Group ◽  
Unitary Operator ◽  
Fock Space ◽  
Adjoint Operator ◽  
The Self ◽  
Canonical Transformations ◽  
Fock Spaces ◽  
Infinitesimal Generators ◽  
Boson Fock Space

A one-parameter symplectic group {etÂ}t∈ℝ derives proper canonical transformations indexed by t on a Boson–Fock space. It has been known that the unitary operator Ut implementing such a proper canonical transformation gives a projective unitary representation of {etÂ}t∈ℝ on the Boson–Fock space and that Ut can be expressed as a normal-ordered form. We rigorously derive the self-adjoint operator Δ(Â) and a local exponent [Formula: see text] with a real-valued function τÂ(·) such that [Formula: see text].

scholarly journals Fourier transforms in generalized Fock spaces

1990 ◽  
Vol 13 (3) ◽  
pp. 431-441
Author(s):  
John Schmeelk
Keyword(s):  
Fourier Transform ◽  
Taylor Series ◽  
Fourier Transforms ◽  
Fock Space ◽  
Natural Generalization ◽  
Fock Spaces ◽  
Test Functions ◽  
Tempered Distributions ◽  
Tempered Distribution ◽  
The Fourier Transform

A classical Fock space consists of functions of the form,Φ↔(ϕ0,ϕ1,…,ϕq,…),whereϕ0∈Candϕq∈L2(R3q),q≥1. We will replace theϕq,q≥1withq-symmetric rapid descent test functions within tempered distribution theory. This space is a natural generalization of a classical Fock space as seen by expanding functionals having generalized Taylor series. The particular coefficients of such series are multilinear functionals having tempered distributions as their domain. The Fourier transform will be introduced into this setting. A theorem will be proven relating the convergence of the transform to the parameter,s, which sweeps out a scale of generalized Fock spaces.

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.

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