scholarly journals Real algebras with a Hilbert space structure

1966 ◽  
Vol 6 (4-5) ◽  
pp. 459-465
Author(s):  
Lars Ingelstam
Keyword(s):  
Hilbert Space ◽  
Space Structure ◽  
Real Algebras

scholarly journals A simple characterization of commutativeH*-algebras

1999 ◽  
Vol 22 (4) ◽  
pp. 885-888
Author(s):  
Parfeny P. Saworotnow
Keyword(s):  
Hilbert Space ◽  
Space Structure ◽  
Simple Characterization

CommutativeH*-algebras are characterized without postulating the existence of Hilbert space structure.

scholarly journals Hilbert space structure and positive operators

2005 ◽  
Vol 305 (2) ◽  
pp. 560-565 ◽  
Author(s):  
Dimosthenis Drivaliaris ◽  
Nikos Yannakakis
Keyword(s):  
Hilbert Space ◽  
Space Structure ◽  
Positive Operators

scholarly journals THE SOq(N, R)-SYMMETRIC HARMONIC OSCILLATOR ON THE QUANTUM EUCLIDEAN SPACE ${\bf R}_q^N$ AND ITS HILBERT SPACE STRUCTURE

1993 ◽  
Vol 08 (26) ◽  
pp. 4679-4729 ◽  
Author(s):  
GAETANO FIORE
Keyword(s):  
Hilbert Space ◽  
Euclidean Space ◽  
Harmonic Oscillator ◽  
Mechanical Model ◽  
Scalar Product ◽  
Space Structure ◽  
Quantum Mechanical ◽  
Quantum Space ◽  
Isotropic Harmonic Oscillator ◽  
Definition Of

We show that the isotropic harmonic oscillator in the ordinary Euclidean space RN (N≥3) admits a natural q-deformation into a new quantum-mechanical model having a q-deformed symmetry (in the sense of quantum groups), SO q(N, R). The q-deformation is the consequence of replacing RN by [Formula: see text] (the corresponding quantum space). This provides an example of quantum mechanics on a noncommutative geometrical space. To reach the goal, we also have to deal with a sensible definition of integration over [Formula: see text], which we use for the definition of the scalar product of states.

scholarly journals The Hilbert-Space Structure of Non-Hermitian Theories with Real Spectra

2004 ◽  
Vol 54 (1) ◽  
pp. 71-75 ◽  
Author(s):  
Ralph Kretschmer ◽  
Lech Szymanowski
Keyword(s):  
Hilbert Space ◽  
Space Structure

scholarly journals Operators on Anti-dual pairs: Self-adjoint Extensions and the Strong Parrott Theorem

2020 ◽  
Vol 63 (4) ◽  
pp. 813-824
Author(s):  
Zsigmond Tarcsay ◽  
Tamás Titkos
Keyword(s):  
Hilbert Space ◽  
Left Ideal ◽  
Space Structure ◽  
Linear Functionals ◽  
Dual Pairs ◽  
Direct Generalization ◽  
Noncommutative Integration ◽  
Block Operator ◽  
Symmetric Operators

AbstractThe aim of this paper is to develop an approach to obtain self-adjoint extensions of symmetric operators acting on anti-dual pairs. The main advantage of such a result is that it can be applied for structures not carrying a Hilbert space structure or a normable topology. In fact, we will show how hermitian extensions of linear functionals of involutive algebras can be governed by means of their induced operators. As an operator theoretic application, we provide a direct generalization of Parrott’s theorem on contractive completion of 2 by 2 block operator-valued matrices. To exhibit the applicability in noncommutative integration, we characterize hermitian extendibility of symmetric functionals defined on a left ideal of a $C^{\ast }$-algebra.

scholarly journals Differential representations of dynamical oscillator symmetries in discrete Hilbert space

2000 ◽  
Vol 5 (2) ◽  
pp. 97-106
Author(s):  
Andreas Ruffing
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Discrete Spectrum ◽  
Classical Limit ◽  
Heisenberg Algebra ◽  
Space Structure ◽  
Hermite Functions ◽  
Dynamical Symmetries ◽  
Creation And Annihilation Operators

As a very important example for dynamical symmetries in the context ofq-generalized quantum mechanics the algebraaa†−q−2a†a=1is investigated. It represents the oscillator symmetrySUq(1,1)and is regarded as a commutation phenomenon of theq-Heisenberg algebra which provides a discrete spectrum of momentum and space,i.e., a discrete Hilbert space structure. Generalizedq-Hermite functions and systems of creation and annihilation operators are derived. The classical limitq→1is investigated. Finally theSUq(1,1)algebra is represented by the dynamical variables of theq-Heisenberg algebra.

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