scholarly journals Commutator criteria for strong mixing

2015 ◽  
Vol 37 (1) ◽  
pp. 308-323 ◽  
Author(s):  
R. TIEDRA DE ALDECOA
Keyword(s):  
Hilbert Space ◽  
General Definition ◽  
Strong Mixing ◽  
Unitary Operators ◽  
Compact Lie Groups ◽  
Skew Products ◽  
Time Changes ◽  
Adjoint Operators ◽  
New Criteria ◽  
Mixing Property

We present new criteria, based on commutator methods, for the strong mixing property of discrete flows $\{U^{N}\}_{N\in \mathbb{Z}}$ and continuous flows $\{e^{-itH}\}_{t\in \mathbb{R}}$ induced by unitary operators $U$ and self-adjoint operators $H$ in a Hilbert space ${\mathcal{H}}$. Our approach put into light a general definition for the topological degree of the maps $N\mapsto U^{N}$ and $t\mapsto e^{-itH}$ with values in the unitary group of ${\mathcal{H}}$. Among other examples, our results apply to skew products of compact Lie groups, time changes of horocycle flows and adjacency operators on graphs.

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 Commutator methods for the spectral analysis of uniquely ergodic dynamical systems

2014 ◽  
Vol 35 (3) ◽  
pp. 944-967 ◽  
Author(s):  
R. TIEDRA DE ALDECOA
Keyword(s):  
Spectral Analysis ◽  
Dynamical Systems ◽  
Absolute Continuity ◽  
Unitary Operators ◽  
Skew Products ◽  
Lebesgue Spectrum ◽  
The Absolute ◽  
Time Changes ◽  
Continuity Of The Spectrum ◽  
Uniquely Ergodic

AbstractWe present a method, based on commutator methods, for the spectral analysis of uniquely ergodic dynamical systems. When applicable, it leads to the absolute continuity of the spectrum of the corresponding unitary operators. As an illustration, we consider time changes of horocycle flows, skew products over translations and Furstenberg transformations. For time changes of horocycle flows we obtain absolute continuity under assumptions weaker than those to be found in the literature, and for skew products over translations and Furstenberg transformations we obtain countable Lebesgue spectrum under assumptions not previously covered in the literature.

A note on normal operators

1963 ◽  
Vol 59 (4) ◽  
pp. 727-729 ◽  
Author(s):  
S. J. Bernau ◽  
F. Smithies
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Adjoint Operator ◽  
Spectral Theorem ◽  
Unitary Operators ◽  
Unitary Space ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Normal Operators ◽  
Finite Dimensional Case

We recall that a bounded linear operator T in a Hilbert space or finite-dimensional unitary space is said to be normal if T commutes with its adjoint operator T*, i.e. TT* = T*T. Most of the proofs given in the literature for the spectral theorem for normal operators, even in the finite-dimensional case, appeal to the corresponding results for Hermitian or unitary operators.

M. A. Krasnosel'skii Theorem and Iterative Methods for Solving Ill-Posed Linear Problems with a Self-Adjoint Operator

2015 ◽  
Vol 15 (3) ◽  
pp. 373-389
Author(s):  
Oleg Matysik ◽  
Petr Zabreiko
Keyword(s):  
Hilbert Space ◽  
Iterative Methods ◽  
Critical Case ◽  
Adjoint Operator ◽  
Successive Approximations ◽  
Operator Equations ◽  
Ill Posed ◽  
Linear Problems ◽  
Adjoint Operators ◽  
Linear Operator Equations

AbstractThe paper deals with iterative methods for solving linear operator equations ${x = Bx + f}$ and ${Ax = f}$ with self-adjoint operators in Hilbert space X in the critical case when ${\rho (B) = 1}$ and ${0 \in \operatorname{Sp} A}$. The results obtained are based on a theorem by M. A. Krasnosel'skii on the convergence of the successive approximations, their modifications and refinements.

scholarly journals Application of Localization to the Multivariate Moment Problem

2014 ◽  
Vol 115 (2) ◽  
pp. 269 ◽  
Author(s):  
Murray Marshall
Keyword(s):  
Hilbert Space ◽  
Moment Problem ◽  
Existence And Uniqueness ◽  
Positive Semidefinite ◽  
Linear Functionals ◽  
Localization Technique ◽  
Adjoint Operators

It is explained how the localization technique introduced by the author in [19] leads to a useful reformulation of the multivariate moment problem in terms of extension of positive semidefinite linear functionals to positive semidefinite linear functionals on the localization of $\mathsf{R}[\underline{x}]$ at $p = \prod_{i=1}^n(1+x_i^2)$ or $p' = \prod_{i=1}^{n-1}(1+x_i^2)$. It is explained how this reformulation can be exploited to prove new results concerning existence and uniqueness of the measure $\mu$ and density of $\mathsf{C}[\underline{x}]$ in $\mathscr{L}^s(\mu)$ and, at the same time, to give new proofs of old results of Fuglede [11], Nussbaum [21], Petersen [22] and Schmüdgen [27], results which were proved previously using the theory of strongly commuting self-adjoint operators on Hilbert space.

The Spectra of Semi-Normal Singular Integral Operators

1970 ◽  
Vol 22 (1) ◽  
pp. 134-150 ◽  
Author(s):  
C. R. Putnam
Keyword(s):  
Hilbert Space ◽  
Singular Integral ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Cauchy Principal ◽  
Cauchy Principal Value ◽  
Principal Value ◽  
Image Position ◽  
Adjoint Operators

Suppose that(1.1)and define the bounded self-adjoint operators H and J on the Hilbert space L2(0, 1) by(1.2)the integral being a Cauchy principal valueIt is seen that(1.3)or, equivalently,(1.4)Since (Cƒ, ƒ) = π–1|(ƒ, ϕ)|2 ≧ 0, A is semi-normal. (For a discussion of such operators, see [4].)

scholarly journals Solution Estimates for the Discrete Lyapunov Equation in a Hilbert Space and Applications to Difference Equations

Axioms ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 20 ◽  
Author(s):  
Michael Gil’
Keyword(s):  
Hilbert Space ◽  
Difference Equation ◽  
Difference Equations ◽  
Lyapunov Equation ◽  
Stability Test ◽  
Norm Estimates ◽  
Point Estimates ◽  
Adjoint Operators ◽  
Solution Estimates

The paper is devoted to the discrete Lyapunov equation X - A * X A = C , where A and C are given operators in a Hilbert space H and X should be found. We derive norm estimates for solutions of that equation in the case of unstable operator A, as well as refine the previously-published estimates for the equation with a stable operator. By the point estimates, we establish explicit conditions, under which a linear nonautonomous difference equation in H is dichotomic. In addition, we suggest a stability test for a class of nonlinear nonautonomous difference equations in H . Our results are based on the norm estimates for powers and resolvents of non-self-adjoint operators.

scholarly journals Spectral Type of One-Parameter Group of Unitary Operators with Transversal Group

1968 ◽  
Vol 32 ◽  
pp. 141-153 ◽  
Author(s):  
Masasi Kowada
Keyword(s):  
Hilbert Space ◽  
Probability Measure ◽  
Spectral Type ◽  
Measure Space ◽  
Unitary Operator ◽  
Separable Hilbert Space ◽  
Unitary Operators ◽  
Parameter Group ◽  
Probability Measure Space

It is an important problem to determine the spectral type of automorphisms or flows on a probability measure space. We shall deal with a unitary operator U and a 1-parameter group of unitary operators {Ut} on a separable Hilbert space H, and discuss their spectral types, although U and {Ut} are not necessarily supposed to be derived from an automorphism or a flow respectively.

On Self-Adjoint Factorization of Operators

1969 ◽  
Vol 21 ◽  
pp. 1421-1426 ◽  
Author(s):  
Heydar Radjavi
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Unitary Operator ◽  
Normal Operator ◽  
Complex Hilbert Space ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Normal Operators ◽  
Adjoint Operators

The main result of this paper is that every normal operator on an infinitedimensional (complex) Hilbert space ℋ is the product of four self-adjoint operators; our Theorem 4 is an actually stronger result. A large class of normal operators will be given which cannot be expressed as the product of three self-adjoint operators.This work was motivated by a well-known resul t of Halmos and Kakutani (3) that every unitary operator on ℋ is the product of four symmetries, i.e., operators that are self-adjoint and unitary.1. By “operator” we shall mean bounded linear operator. The space ℋ will be infinite-dimensional (separable or non-separable) unless otherwise specified. We shall denote the class of self-adjoint operators on ℋ by and that of symmetries by .

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