CONTRACTIONS OF LIE ALGEBRA REPRESENTATIONS

1999 ◽  
Vol 11 (10) ◽  
pp. 1179-1207 ◽  
Author(s):  
U. CATTANEO ◽  
W. F. WRESZINSKI
Keyword(s):  
General Theory ◽  
Banach Spaces ◽  
Lie Algebra ◽  
Hilbert Spaces ◽  
3 Dimensional ◽  
Algebra Representation ◽  
Carrier Space ◽  
Definition Of ◽  
Lie Algebra Representations ◽  
Main Distinguishing Feature

A theory of contractions of Lie algebra representations on complex Hilbert spaces is proposed, based on Trotter's theory of approximating sequences of Banach spaces. Its main distinguishing feature is a careful definition of the carrier space of the limit Lie algebra representation. A set of quite general conditions on the contracting representations, satisfied in all known examples, is proven to be sufficient for the existence of such a representation. In order to show how natural the suggested framework is, the general theory is applied to the contraction of [Formula: see text] into the Lie algebra [Formula: see text] of the 3-dimensional Heisenberg group and to the related study of the limit N→∞ of a quantum system of N identical two-level particles.

scholarly journals LIE ALGEBRA REPRESENTATIONS AND RIGGED HILBERT SPACES: THE SO(2) CASE

2017 ◽  
Vol 57 (6) ◽  
pp. 379 ◽  
Author(s):  
Enrico Celeghini ◽  
Manuel Gadella ◽  
Mariano A Del Olmo
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Hilbert Spaces ◽  
Irreducible Representations ◽  
Rigged Hilbert Spaces ◽  
Suitable Framework ◽  
Lie Algebra Representations

It is well known that related with the irreducible representations of the Lie group <em>SO</em>(2) we find a discrete basis as well a continuous one. In this paper we revisited this situation under the light of  Rigged Hilbert spaces, which are the suitable framework to deal  with both discrete and bases in the same context and in  relation with physical applications.

scholarly journals On a second numerical index for Banach spaces

2019 ◽  
Vol 150 (2) ◽  
pp. 1003-1051 ◽  
Author(s):  
Sun Kwang Kim ◽  
Han Ju Lee ◽  
Miguel Martín ◽  
Javier Merí
Keyword(s):  
Banach Spaces ◽  
Lie Algebra ◽  
Hilbert Spaces ◽  
Unconditional Basis ◽  
Numerical Radius ◽  
Best Constant ◽  
The One ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Numerical Index

AbstractWe introduce a second numerical index for real Banach spaces with non-trivial Lie algebra, as the best constant of equivalence between the numerical radius and the quotient of the operator norm modulo the Lie algebra. We present a number of examples and results concerning absolute sums, duality, vector-valued function spaces…which show that, in many cases, the behaviour of this second numerical index differs from the one of the classical numerical index. As main results, we prove that Hilbert spaces have second numerical index one and that they are the only spaces with this property among the class of Banach spaces with one-unconditional basis and non-trivial Lie algebra. Besides, an application to the Bishop-Phelps-Bollobás property for the numerical radius is given.

Dualism QM and GR

2019 ◽  
Author(s):  
Vitaly Kuyukov
Keyword(s):  
Quantum Mechanics ◽  
General Theory ◽  
Noncommutative Geometry ◽  
Quantum Tunneling ◽  
Closed Surface ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Surface Integral ◽  
Definition Of

Quantum tunneling of noncommutative geometry gives the definition of time in the form of holography, that is, in the form of a closed surface integral. Ultimately, the holography of time shows the dualism between quantum mechanics and the general theory of relativity.

scholarly journals Existence and optimal controls for nonlocal fractional evolution equations of order (1,2) in Banach spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Denghao Pang ◽  
Wei Jiang ◽  
Azmat Ullah Khan Niazi ◽  
Jiale Sheng
Keyword(s):  
Banach Spaces ◽  
Mild Solution ◽  
Measure Of Noncompactness ◽  
Evolution Equations ◽  
Well Posedness ◽  
Optimal Controls ◽  
Lagrange Problem ◽  
Fractional Differential ◽  
Fractional Evolution Systems ◽  
Definition Of

AbstractIn this paper, we mainly investigate the existence, continuous dependence, and the optimal control for nonlocal fractional differential evolution equations of order (1,2) in Banach spaces. We define a competent definition of a mild solution. On this basis, we verify the well-posedness of the mild solution. Meanwhile, with a construction of Lagrange problem, we elaborate the existence of optimal pairs of the fractional evolution systems. The main tools are the fractional calculus, cosine family, multivalued analysis, measure of noncompactness method, and fixed point theorem. Finally, an example is propounded to illustrate the validity of our main results.

scholarly journals Sets of periods for chaotic linear operators

2021 ◽  
Vol 115 (2) ◽  
Author(s):  
J. A. Conejero ◽  
F. Martínez-Giménez ◽  
A. Peris ◽  
F. Rodenas
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Complete Characterization ◽  
Linear Operators

AbstractWe provide a complete characterization of the possible sets of periods for Devaney chaotic linear operators on Hilbert spaces. As a consequence, we also derive this characterization for linearizable maps on Banach spaces.

scholarly journals A characterisation of Hilbert spaces via orthogonality and proximinality

2005 ◽  
Vol 71 (1) ◽  
pp. 107-111
Author(s):  
Fathi B. Saidi
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Real Banach Space ◽  
Nonzero Element ◽  
Dimensional Subspace ◽  
Two Dimensional ◽  
Orthogonal Direction ◽  
Open Problems

In this paper we adopt the notion of orthogonality in Banach spaces introduced by the author in [6]. There, the author showed that in any two-dimensional subspace F of E, every nonzero element admits at most one orthogonal direction. The problem of existence of such orthogonal direction was not addressed before. Our main purpose in this paper is the investigation of this problem in the case where E is a real Banach space. As a result we obtain a characterisation of Hilbert spaces stating that, if in every two-dimensional subspace F of E every nonzero element admits an orthogonal direction, then E is isometric to a Hilbert space. We conclude by presenting some open problems.

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