scholarly journals IMPLEMENTATION OF ENDOMORPHISMS OF THE CAR ALGEBRA

1995 ◽  
Vol 07 (06) ◽  
pp. 833-869 ◽  
Author(s):  
CARSTEN BINNENHEI
Keyword(s):  
Hilbert Space ◽  
Connected Components ◽  
Square Root ◽  
Fredholm Index ◽  
Fock States ◽  
Car Algebra ◽  
Bogoliubov Transformations ◽  
Particle Operator ◽  
Explicit Expressions

The implementation of non-surjective Bogoliubov transformations in Fock states over CAR algebras is investigated. Such a transformation is implementable by a Hilbert space of isometries if and only if the well-known Shale-Stinespring condition is met. In this case, the dimension of the implementing Hilbert space equals the square root of the Watatani index of the associated inclusion of CAR algebras, and both are determined by the Fredholm index of the corresponding one-particle operator. Explicit expressions for the implementing operators are obtained, and the connected components of the semigroup of implementable transformations are described.

scholarly journals Decay of correlations in suspension semi-flows of angle-multiplying maps

2008 ◽  
Vol 28 (1) ◽  
pp. 291-317 ◽  
Author(s):  
MASATO TSUJII
Keyword(s):  
Hilbert Space ◽  
Sobolev Space ◽  
Spectral Radius ◽  
Decay Of Correlations ◽  
Square Root ◽  
Precise Description ◽  
Anisotropic Sobolev Space ◽  
Essential Spectral Radius ◽  
Frobenius Operator ◽  
Dense Set

AbstractWe consider suspension semi-flows of angle-multiplying maps on the circle for Cr ceiling functions with r≥3. Under a Crgeneric condition on the ceiling function, we show that there exists a Hilbert space (anisotropic Sobolev space) contained in the L2 space such that the Perron–Frobenius operator for the time-t-map acts naturally on it and that the essential spectral radius of that action is bounded by the square root of the inverse of the minimum expansion rate. This leads to a precise description of decay of correlations. Furthermore, the Perron–Frobenius operator for the time-t-map is quasi-compact for a Cr open and dense set of ceiling functions.

scholarly journals Determinants, Pfaffians and Quasi-Free Representations of the CAR Algebra

1998 ◽  
Vol 10 (05) ◽  
pp. 705-721 ◽  
Author(s):  
Mauro Spera ◽  
Tilmann Wurzbacher
Keyword(s):  
Hilbert Space ◽  
Line Bundle ◽  
Line Bundles ◽  
Complex Structures ◽  
Square Root ◽  
Purification Procedure ◽  
Algebraic Construction ◽  
Infinite Dimensional ◽  
Weil Type ◽  
Determinant Line Bundle

In this paper we apply the theory of quasi-free states of CAR algebras and Bogolubov automorphisms to give an alternative C*-algebraic construction of the Determinant and Pfaffian line bundles discussed by Pressley and Segal and by Borthwick. The basic property of the Pfaffian of being the holomorphic square root of the Determinant line bundle (after restriction from the Hilbert space Grassmannian to the Siegel manifold, or isotropic Grassmannian, consisting of all complex structures on an associated Hilbert space) is derived from a Fock–anti-Fock correspondence and an application of the Powers–Størmer purification procedure. A Borel–Weil type description of the infinite dimensional Spin c- representation is obtained, via a Shale–Stinespring implementation of Bogolubov transformations.

scholarly journals Lipschitz Extensions to Finitely Many Points

2018 ◽  
Vol 6 (1) ◽  
pp. 174-191 ◽  
Author(s):  
Giuliano Basso
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Metric Spaces ◽  
Lipschitz Constant ◽  
Target Space ◽  
Square Root ◽  
Lipschitz Maps ◽  
Lipschitz Map ◽  
Lipschitz Extensions

AbstractWe consider Lipschitz maps with values in quasi-metric spaces and extend such maps to finitely many points. We prove that in this context every 1-Lipschitz map admits an extension such that its Lipschitz constant is bounded from above by the number of added points plus one. Moreover, we prove that if the source space is a Hilbert space and the target space is a Banach space, then there exists an extension such that its Lipschitz constant is bounded from above by the square root of the total of added points plus one. We discuss applications to metric transforms.

Tensor Product Decomposition, Entanglement, and Bogoliubov Transformations for Two Fermion System

2005 ◽  
Vol 12 (02) ◽  
pp. 179-188 ◽  
Author(s):  
Paweł Caban ◽  
Krzysztof Podlaski ◽  
Jakub Rembieliński ◽  
Kordian A. Smoliński ◽  
Zbigniew Walczak
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Explicit Formula ◽  
Fermion System ◽  
Density Matrices ◽  
Product Decomposition ◽  
Creation And Annihilation Operators ◽  
Separable States ◽  
Entanglement Of Formation ◽  
Bogoliubov Transformations

We consider the two-fermion system whose states are subjected to the superselection rule forbidding the superposition of states with fermionic and bosonic statistics. This implies that separable states are described only by diagonal density matrices. Moreover, we find the explicit formula for the entanglement of formation, which in this case cannot be calculated properly using Wootters's concurrence. We also discuss the problem of the choice of tensor product decomposition in a system of two fermions with the help of Bogoliubov transformations of creation and annihilation operators. Finally, we show that there exist states which are separable with respect to all tensor product decompositions of the underlying Hilbert space.

SOBOLEV REGULARITY OF INVARIANT MEASURES FOR GENERALIZED ORNSTEIN–UHLENBECK OPERATORS

2006 ◽  
Vol 09 (04) ◽  
pp. 595-606 ◽  
Author(s):  
ABDELHADI ES-SARHIR
Keyword(s):  
Hilbert Space ◽  
Invariant Measure ◽  
Sobolev Space ◽  
Invariant Measures ◽  
Separable Hilbert Space ◽  
Square Root ◽  
Absolutely Continuous ◽  
Sobolev Regularity ◽  
Reference Measure ◽  
Nikodym Derivative

This paper deals with the regularity of an invariant measure μ associated to a class of generalized Ornstein–Uhlenbeck operators. Regularity here means that μ is absolutely continuous with respect to a properly chosen Gaussian reference measure σ on a separable Hilbert space H. Moreover, the square root of its Radon–Nikodym derivative ρ should belong to some directional Sobolev space [Formula: see text].

Characteristics of the Elastic Field of a Rigid Line Inhomogeneity

1985 ◽  
Vol 52 (4) ◽  
pp. 818-822 ◽  
Author(s):  
Z. Y. Wang ◽  
H. T. Zhang ◽  
Y. T. Chou
Keyword(s):  
Stress Field ◽  
Elastic Field ◽  
Square Root ◽  
Displacement Fields ◽  
Stress And Displacement Fields ◽  
Rigid Line ◽  
Square Root Singularity ◽  
Inclined Loading ◽  
Explicit Expressions

Explicit expressions are obtained for the stress and displacement fields near the tip of a rigid line inhomogeneity subjected to an inclined loading. It is shown that the tip stress field, with a square-root singularity, differs in characteristics from that of a slit crack. New designations for the mode of deformation based on the fracture concept are presented and the inhomogeneity extension forces are calculated and discussed.

scholarly journals ON THE COMPONENTS OF SPACES OF CURVES ON THE 2-SPHERE WITH GEODESIC CURVATURE IN A PRESCRIBED INTERVAL

2013 ◽  
Vol 24 (14) ◽  
pp. 1350101 ◽  
Author(s):  
NICOLAU C. SALDANHA ◽  
PEDRO ZÜHLKE
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Geodesic Curvature ◽  
Connected Components ◽  
Closed Curves ◽  
Simple Characterization ◽  
Class C

Let [Formula: see text] denote the set of all closed curves of class Cron the sphere S2whose geodesic curvatures are constrained to lie in (κ1, κ2), furnished with the Crtopology (for some r ≥ 2 and possibly infinite κ1< κ2). In 1970, J. Li ttle proved that the space [Formula: see text] of closed curves having positive geodesic curvature has three connected components. Let ρi= arccot κi(i = 1, 2). We show that [Formula: see text] has n connected components [Formula: see text] where [Formula: see text] and [Formula: see text] contains circles traversed j times (1 ≤ j ≤ n). The component [Formula: see text] also contains circles traversed (n - 1) + 2k times, and [Formula: see text] also contains circles traversed n + 2k times, for any k ∈ N. Further, each of [Formula: see text](n ≥ 3) is homeomorphic to SO3× E, where E is the separable Hilbert space. We also obtain a simple characterization of the components in terms of the properties of a curve and prove that [Formula: see text] is homeomorphic to [Formula: see text] whenever [Formula: see text].

scholarly journals On Square Roots and Logarithms of Self-Adjoint Operators

1958 ◽  
Vol 4 (1) ◽  
pp. 1-2 ◽  
Author(s):  
C. R. Putnam
Keyword(s):  
Hilbert Space ◽  
Sufficient Condition ◽  
Square Root ◽  
The Real ◽  
Square Roots ◽  
Simple Sufficient Condition ◽  
Image Position ◽  
Adjoint Operators

All operators considered in this paper are bounded and linear (everywhere defined) on a Hilbert space. An operator A will be called a square root of an operator B ifA simple sufficient condition guaranteeing that any solution A of (1) be normal whenever B is normal was obtained in [1], namely: If B is normal and if there exists some real angle θ for which Re(Aeιθ)≥0, then (1) implies that A is normal. Here, Re (C) denotes the real part ½(C + C*) of an operator C.

scholarly journals Self-adjoint square roots of positive self-adjoint bounded linear operators

1972 ◽  
Vol 18 (1) ◽  
pp. 77-79 ◽  
Author(s):  
C. A. Stuart
Keyword(s):  
Hilbert Space ◽  
Linear Operators ◽  
Square Root ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Linear Map ◽  
Square Roots

A corollary of the main theorem presented in this note is a generalisation of the well-known result that a self-adjoint square root of a positive self-adjoint compact linear map in a Hilbert space is itself a compact linear map. The method used here exploits the techniques developed recently in the study of k-set contractions ((1), (2)).

scholarly journals The JIMWLK evolution and the s-channel unitarity

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Alex Kovner ◽  
Eugene Levin ◽  
Ming Li ◽  
Michael Lublinsky
Keyword(s):  
Field Theory ◽  
Hilbert Space ◽  
Scattering Amplitudes ◽  
Degrees Of Freedom ◽  
Energy Evolution ◽  
Fock States ◽  
Target States ◽  
Reggeon Field Theory ◽  
Dilute Limit

Abstract Further developing ideas set forth in [1], we discuss QCD Reggeon Field Theory (RFT) and formulate restrictions imposed on its Hamiltonian by the unitarity of underlying QCD. We identify explicitly the QCD RFT Hilbert space, provide algebra of the basic degrees of freedom (Wilson lines and their duals) and the algorithm for calculating the scattering amplitudes. We formulate conditions imposed on the “Fock states” of RFT by unitary nature of QCD, and explain how these constraints appear as unitarity constraints on possible RFT hamiltonians that generate energy evolution of scattering amplitudes. We study the realization of these constraints in the dense-dilute limit of RFT where the appropriate Hamiltonian is the JIMWLK Hamiltonian HJIMWLK. We find that the action HJIMWLK on the dilute projectile states is unitary, but acting on dense “target” states it violates unitarity and generates states with negative probabilities through energy evolution.

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