Approximately transitive (2) flows and transformations have simple spectrum

1988 ◽  
pp. 261-280 ◽  
Author(s):  
Jane M. Hawkins ◽  
E. A. Robinson
Keyword(s):  
Simple Spectrum

Ergodic transformations conjugate to their inverses by involutions

1996 ◽  
Vol 16 (1) ◽  
pp. 97-124 ◽  
Author(s):  
Geoffrey R. Goodson ◽  
Andrés del Junco ◽  
Mariusz Lemańczyk ◽  
Daniel J. Rudolph
Keyword(s):  
Discrete Spectrum ◽  
Probability Space ◽  
Closure Property ◽  
Simple Spectrum ◽  
Ergodic Transformations ◽  
Ergodic Automorphism ◽  
Singular Spectrum ◽  
Weak Closure ◽  
Identity Automorphism ◽  
Borel Probability

AbstractLetTbe an ergodic automorphism defined on a standard Borel probability space for whichTandT−1are isomorphic. We investigate the form of the conjugating automorphism. It is well known that ifTis ergodic having a discrete spectrum andSis the conjugation betweenTandT−1, i.e.SsatisfiesTS=ST−1thenS2=Ithe identity automorphism. We show that this result remains true under the weaker assumption thatThas a simple spectrum. IfThas the weak closure property and is isomorphic to its inverse, it is shown that the conjugationSsatisfiesS4=I. Finally, we construct an example to show that the conjugation need not be an involution in this case. The example we construct, in addition to having the weak closure property, is of rank two, rigid and simple for all orders with a singular spectrum of multiplicity equal to two.

scholarly journals Shift invariant measures and simple spectrum

2000 ◽  
Vol 84 (2) ◽  
pp. 385-394 ◽  
Author(s):  
A. Kłopotowski ◽  
M. Nadkarni
Keyword(s):  
Invariant Measures ◽  
Simple Spectrum

Simple Spectrum and Rayleigh Quotients

2014 ◽  
pp. 33-44 ◽  
Author(s):  
Victor Guillemin ◽  
Eveline Legendre ◽  
Rosa Sena-Dias
Keyword(s):  
Simple Spectrum

Genericity of Certain Classes of Unitary and Self-Adjoint Operators

1998 ◽  
Vol 41 (2) ◽  
pp. 137-139 ◽  
Author(s):  
J. R. Choksi ◽  
M. G. Nadkarni
Keyword(s):  
Hilbert Space ◽  
Conference Proceedings ◽  
Separable Hilbert Space ◽  
Strong Operator Topology ◽  
Slight Modification ◽  
Simple Spectrum ◽  
Cayley Transform ◽  
Unitary Operators ◽  
Full Support ◽  
Adjoint Operators

AbstractIn a paper [1], published in 1990, in a (somewhat inaccessible) conference proceedings, the authors had shown that for the unitary operators on a separable Hilbert space, endowed with the strong operator topology, those with singular, continuous, simple spectrum, with full support, forma dense Gδ. A similar theoremfor bounded self-adjoint operators with a given normbound (omitting simplicity) was recently given by Barry Simon [2], [3], with a totally different proof. In this note we show that a slight modification of our argument, combined with the Cayley transform, gives a proof of Simon’s result, with simplicity of the spectrum added.

scholarly journals ON SEPARATION OF VARIABLES AND COMPLETENESS OF THE BETHE ANSATZ FOR QUANTUM N GAUDIN MODEL

2009 ◽  
Vol 51 (A) ◽  
pp. 137-145 ◽  
Author(s):  
E. MUKHIN ◽  
V. TARASOV ◽  
A. VARCHENKO
Keyword(s):  
Differential Operator ◽  
Bethe Ansatz ◽  
Differential Operators ◽  
Separation Of Variables ◽  
Polynomial Kernel ◽  
Simple Spectrum ◽  
Gaudin Model ◽  
Bethe Algebra ◽  
One To One ◽  
Evaluation Parameters

AbstractIn this paper, we discuss implications of the results obtained in [5]. It was shown there that eigenvectors of the Bethe algebra of the quantum N Gaudin model are in a one-to-one correspondence with Fuchsian differential operators with polynomial kernel. Here, we interpret this fact as a separation of variables in the N Gaudin model. Having a Fuchsian differential operator with polynomial kernel, we construct the corresponding eigenvector of the Bethe algebra. It was shown in [5] that the Bethe algebra has simple spectrum if the evaluation parameters of the Gaudin model are generic. In that case, our Bethe ansatz construction produces an eigenbasis of the Bethe algebra.

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