scholarly journals Transfer operators for ultradifferentiable expanding maps of the circle

2020 ◽  
pp. 1-20 ◽  
Author(s):  
MALO JÉZÉQUEL
Keyword(s):  
Hilbert Space ◽  
Compact Operator ◽  
Nuclear Power ◽  
Transfer Operator ◽  
Singular Values ◽  
Expanding Maps ◽  
Smooth Functions ◽  
Transfer Operators ◽  
Expanding Map

Given a ${\mathcal{C}}^{\infty }$ expanding map $T$ of the circle, we construct a Hilbert space ${\mathcal{H}}$ of smooth functions on which the transfer operator ${\mathcal{L}}$ associated to $T$ acts as a compact operator. This result is made quantitative (in terms of singular values of the operator ${\mathcal{L}}$ acting on ${\mathcal{H}}$ ) using the language of Denjoy–Carleman classes. Moreover, the nuclear power decomposition of Baladi and Tsujii can be performed on the space ${\mathcal{H}}$ , providing a bound on the growth of the dynamical determinant associated to ${\mathcal{L}}$ .

scholarly journals Spectral theory of discrete processes

Open Physics ◽  
2010 ◽  
Vol 8 (3) ◽  
Author(s):  
Palle Jorgensen ◽  
Myung-Sin Song
Keyword(s):  
Stochastic Process ◽  
Hilbert Space ◽  
Spectral Analysis ◽  
Stochastic Processes ◽  
Spectral Theory ◽  
Transfer Operator ◽  
Infinite Graphs ◽  
Spectral Theorem ◽  
Transfer Operators ◽  
Wide Range

AbstractWe offer a spectral analysis for a class of transfer operators. These transfer operators arise for a wide range of stochastic processes, ranging from random walks on infinite graphs to the processes that govern signals and recursive wavelet algorithms; even spectral theory for fractal measures. In each case, there is an associated class of harmonic functions which we study. And in addition, we study three questions in depthIn specific applications, and for a specific stochastic process, how do we realize the transfer operator T as an operator in a suitable Hilbert space? And how to spectral analyze T once the right Hilbert space H has been selected? Finally we characterize the stochastic processes that are governed by a single transfer operator.In our applications, the particular stochastic process will live on an infinite path-space which is realized in turn on a state space S. In the case of random walk on graphs G, S will be the set of vertices of G. The Hilbert space H on which the transfer operator T acts will then be an L 2 space on S, or a Hilbert space defined from an energy-quadratic form.This circle of problems is both interesting and non-trivial as it turns out that T may often be an unbounded linear operator in H; but even if it is bounded, it is a non-normal operator, so its spectral theory is not amenable to an analysis with the use of von Neumann’s spectral theorem. While we offer a number of applications, we believe that our spectral analysis will have intrinsic interest for the theory of operators in Hilbert space.

Inequalities for eigenvalues of compact operators in a Hilbert space

2016 ◽  
Vol 18 (01) ◽  
pp. 1550022 ◽  
Author(s):  
Michael Gil’
Keyword(s):  
Hilbert Space ◽  
Compact Operator ◽  
Separable Hilbert Space ◽  
Singular Values ◽  
Compact Operators ◽  
Classical Inequality

Let [Formula: see text] be a compact operator in a separable Hilbert space and [Formula: see text] be the eigenvalues of [Formula: see text] with their multiplicities enumerated in the non-increasing order of their absolute values. We prove the inequality [Formula: see text] where [Formula: see text] and [Formula: see text] are the singular values of [Formula: see text] and of [Formula: see text], respectively, enumerated with their multiplicities in the non-increasing order. This result refines the classical inequality [Formula: see text]

scholarly journals Explicit coupling argument for non-uniformly hyperbolic transformations

2018 ◽  
Vol 149 (1) ◽  
pp. 101-130 ◽  
Author(s):  
A. Korepanov ◽  
Z. Kosloff ◽  
I. Melbourne
Keyword(s):  
Spectral Properties ◽  
Transfer Operator ◽  
Decay Of Correlations ◽  
Exponential Polynomial ◽  
Expanding Maps ◽  
Stretched Exponential ◽  
Expanding Map ◽  
Mixing Rates ◽  
Rates Of Decay

The transfer operator corresponding to a uniformly expanding map enjoys good spectral properties. We verify that coupling yields explicit estimates that depend continuously on the expansion and distortion constants of the map. For non-uniformly expanding maps with a uniformly expanding induced map, we obtain explicit estimates for mixing rates (exponential, stretched exponential, polynomial) that again depend continuously on the constants for the induced map together with data associated with the inducing time. Finally, for non-uniformly hyperbolic transformations, we obtain the corresponding estimates for rates of decay of correlations.

The nuclearity of operators generated by Hölder continuous kernels

1974 ◽  
Vol 75 (3) ◽  
pp. 351-356 ◽  
Author(s):  
James Alan Cochran
Keyword(s):  
Hilbert Space ◽  
Compact Operator ◽  
Singular Values ◽  
Trace Class ◽  
Hölder Continuous ◽  
Nuclear Operators ◽  
Characteristic Values ◽  
Image Position

Introduction. The space Cp is the class of operators on a Hilbert space for which the norm ∥K∥p = [trace (KK*)P/2]/p is finite. Equivalently, a compact operator is in Cp ifwhere the μn are the so-called ‘singular values’ of K (characteristic values of the non-negative compact operator [K] ≡ (KK*)½). The case p = 2 gives the familiar class of Hilbert–Schmidt operators, while C1 is the collection of trace-class or nuclear operators considered by Schatten(12), Lidskii(11), and Gohberg and Krein(7), among others.

scholarly journals On Gibbs Measures and Spectra of Ruelle Transfer Operators

2017 ◽  
Vol 60 (2) ◽  
pp. 411-421
Author(s):  
Luchezar Stoyanov
Keyword(s):  
Spectral Radius ◽  
Spectral Properties ◽  
Gibbs Measures ◽  
Transfer Operator ◽  
Frobenius Theorem ◽  
Transfer Operators

AbstractWe prove a comprehensive version of the Ruelle–Perron–Frobenius Theorem with explicit estimates of the spectral radius of the Ruelle transfer operator and various other quantities related to spectral properties of this operator. The novelty here is that the Hölder constant of the function generating the operator appears only polynomially, not exponentially as in previously known estimates.

Weyl's theorem for the square of operator and perturbations

2015 ◽  
Vol 17 (05) ◽  
pp. 1450042
Author(s):  
Weijuan Shi ◽  
Xiaohong Cao
Keyword(s):  
Hilbert Space ◽  
Compact Operator ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Weyl’S Theorem ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Weyl Spectrum ◽  
The Stability

Let H be an infinite-dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. T ∈ B(H) satisfies Weyl's theorem if σ(T)\σw(T) = π00(T), where σ(T) and σw(T) denote the spectrum and the Weyl spectrum of T, respectively, π00(T) = {λ ∈ iso σ(T) : 0 < dim N(T - λI) < ∞}. T ∈ B(H) is said to have the stability of Weyl's theorem if T + K satisfies Weyl's theorem for all compact operator K ∈ B(H). In this paper, we characterize the operator T on H satisfying the stability of Weyl's theorem holds for T2.

A contribution to the solution of the compact correction problem for operators on a Banach space

1989 ◽  
Vol 31 (2) ◽  
pp. 219-229
Author(s):  
Mícheál Ó Searcóid
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Compact Operator ◽  
Complex Number ◽  
Space Problem ◽  
Riesz Operators ◽  
Correction Problem ◽  
Bounded Below ◽  
General Banach Space

We consider the hypothesis that an operator T on a given Banach space can always be perturbed by a compact operator K in such a way that, whenever a complex number A is in the semi-Fredholm region of T + K, then T + K – λ is either bounded below or surjective. The hypothesis has its origin in the work of West [11], who proved it for Riesz operators on Hilbert space. In this paper, we reduce the general Banach space problem to one of considering only operators of a special type, operators which are, in a spectral sense, natural generalizations of the Riesz operators studied by West.

Sign in / Sign up

Export Citation Format

Share Document