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.

The Hodge Operator in Fermionic Fock Space

2005 ◽  
Vol 70 (7) ◽  
pp. 979-1016 ◽  
Author(s):  
Leszek Z. Stolarczyk
Keyword(s):  
Fock Space ◽  
Differential Forms ◽  
Grassmann Algebra ◽  
Basis Set ◽  
Electron Systems ◽  
Linear Transformations ◽  
Spin Orbital ◽  
Creation And Annihilation Operators ◽  
Finite Dimensional ◽  
Hodge Operator

The Hodge operator ("star" operator) plays an important role in the theory of differential forms, where it serves as a tool for the switching between the exterior derivative and co-derivative. In the theory of many-electron systems involving a finite-dimensional fermionic Fock space, one can define the Hodge operator as a unique (i.e., invariant with respect to linear transformations of the spin-orbital basis set) antilinear operator. The similarity transformation based on the Hodge operator results in the switching between the fermion creation and annihilation operators. The present paper gives a self-contained account on the algebraic structures which are necessary for the construction of the Hodge operator: the fermionic Fock space, the corresponding Grassmann algebra, and the generalized creation and annihilation operators. The Hodge operator is then defined, and its properties are reviewed. It is shown how the notion of the Hodge operator can be employed in a construction of the electronic time-reversal operator.

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 Lebesgue Decomposition of Non-Commutative Measures

2020 ◽  
Author(s):  
Michael T Jury ◽  
Robert T W Martin
Keyword(s):  
Hilbert Space ◽  
Reproducing Kernel ◽  
Fock Space ◽  
Reproducing Kernel Hilbert Space ◽  
Semigroup Algebra ◽  
Space Theory ◽  
Lebesgue Decomposition ◽  
Sesquilinear Forms ◽  
Positive Linear Functionals ◽  
Algebra Theory

Abstract We extend the Lebesgue decomposition of positive measures with respect to Lebesgue measure on the complex unit circle to the non-commutative (NC) multi-variable setting of (positive) NC measures. These are positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz $C^{\ast }-$algebra, the $C^{\ast }-$algebra of the left creation operators on the full Fock space. This theory is fundamentally connected to the representation theory of the Cuntz and Cuntz–Toeplitz $C^{\ast }-$algebras; any *−representation of the Cuntz–Toeplitz $C^{\ast }-$algebra is obtained (up to unitary equivalence), by applying a Gelfand–Naimark–Segal construction to a positive NC measure. Our approach combines the theory of Lebesgue decomposition of sesquilinear forms in Hilbert space, Lebesgue decomposition of row isometries, free semigroup algebra theory, NC reproducing kernel Hilbert space theory, and NC Hardy space theory.

On Closed Subsets of Root Systems

1994 ◽  
Vol 37 (3) ◽  
pp. 338-345 ◽  
Author(s):  
D. Ž. Doković ◽  
P. Check ◽  
J.-Y. Hée
Keyword(s):  
Weyl Group ◽  
Root System ◽  
Irreducible Component ◽  
Product Space ◽  
Root Systems ◽  
Inner Product Space ◽  
Inner Product ◽  
Closed Sets ◽  
Finite Dimensional ◽  
Real Inner Product Space

AbstractLet R be a root system (in the sense of Bourbaki) in a finite dimensional real inner product space V. A subset P ⊂ R is closed if α, β ∊ P and α + β ∊ R imply that α + β ∊ P. In this paper we shall classify, up to conjugacy by the Weyl group W of R, all closed sets P ⊂ R such that R\P is also closed. We also show that if θ:R —> R′ is a bijection between two root systems such that both θ and θ-1 preserve closed sets, and if R has at most one irreducible component of type A1, then θ is an isomorphism of root systems.

scholarly journals MATRIX THEORY, HILBERT SCHEME AND INTEGRABLE SYSTEM

1998 ◽  
Vol 13 (34) ◽  
pp. 2731-2742 ◽  
Author(s):  
YUTAKA MATSUO
Keyword(s):  
Hilbert Scheme ◽  
Matrix Theory ◽  
Fock Space ◽  
Homology Class ◽  
Orthogonal Basis ◽  
Inner Product ◽  
Virasoro Symmetry ◽  
Hilbert Scheme Of Points ◽  
The Matrix ◽  
Boson Fock Space

We give a reinterpretation of the matrix theory discussed by Moore, Nekrasov and Shatashivili (MNS) in terms of the second quantized operators which describes the homology class of the Hilbert scheme of points on surfaces. It naturally relates the contribution from each pole to the inner product of orthogonal basis of free boson Fock space. These bases can be related to the eigenfunctions of Calogero–Sutherland (CS) equation and the deformation parameter of MNS is identified with coupling of CS system. We discuss the structure of Virasoro symmetry in this model.

scholarly journals A Unified Reproducing Kernel Method and Error Estimation for Solving Linear Differential Equation with Functional Constraints

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Xinjian Zhang ◽  
Xiongwei Liu
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Linear Differential Equation ◽  
Error Estimation ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Inner Product ◽  
Reproducing Kernel Method ◽  
Linear Differential ◽  
Polynomial Reproducing

A unified reproducing kernel method for solving linear differential equations with functional constraint is provided. We use a specified inner product to obtain a class of piecewise polynomial reproducing kernels which have a simple unified description. Arbitrary order linear differential operator is proved to be bounded about the special inner product. Based on space decomposition, we present the expressions of exact solution and approximate solution of linear differential equation by the polynomial reproducing kernel. Error estimation of approximate solution is investigated. Since the approximate solution can be described by polynomials, it is very suitable for numerical calculation.

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 A $\lambda$-ring Frobenius Characteristic for $G\wr S_n$

10.37236/1809 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Anthony Mendes ◽  
Jeffrey Remmel ◽  
Jennifer Wagner
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Kronecker Product ◽  
Reproducing Kernel ◽  
Irreducible Representations ◽  
Inner Product ◽  
Irreducible Components ◽  
Lambda Ring

A $\lambda$-ring version of a Frobenius characteristic for groups of the form $G \wr S_n$ is given. Our methods provide natural analogs of classic results in the representation theory of the symmetric group. Included is a method decompose the Kronecker product of two irreducible representations of $G\wr S_n$ into its irreducible components along with generalizations of the Murnaghan-Nakayama rule, the Hall inner product, and the reproducing kernel for $G\wr S_n$.

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.

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