A Survey on the Behavior of the Resolvent of Weighted Shift Operator

AbstractWe study the stability of linear filters associated with certain types of linear difference equations with variable coefficients. We show that stability is determined by the locations of the poles of a rational transfer function relative to the spectrum of an associated weighted shift operator. The known theory for filters associated with constant-coefficient difference equations is a special case.

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CHAOS IN A CLASS OF NONCONSTANT WEIGHTED SHIFT OPERATORS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127413500107 ◽

2013 ◽

Vol 23 (01) ◽

pp. 1350010 ◽

Cited By ~ 2

Author(s):

XINXING WU ◽

PEIYONG ZHU

Keyword(s):

Positive Integer ◽

Linear Space ◽

Normed Linear Space ◽

Shift Operator ◽

Weighted Shift ◽

Constructive Proof ◽

Sufficient Condition ◽

Weighted Shift Operator ◽

Shift Operators ◽

Devaney Chaos

In this paper, chaos generated by a class of nonconstant weighted shift operators is studied. First, we prove that for the weighted shift operator Bμ : Σ(X) → Σ(X) defined by Bμ(x0, x1, …) = (μ(0)x1, μ(1)x2, …), where X is a normed linear space (not necessarily complete), weak mix, transitivity (hypercyclity) and Devaney chaos are all equivalent to separability of X and this property is preserved under iterations. Then we get that [Formula: see text] is distributionally chaotic and Li–Yorke sensitive for each positive integer N. Meanwhile, a sufficient condition ensuring that a point is k-scrambled for all integers k > 0 is obtained. By using these results, a simple example is given to show that Corollary 3.3 in [Fu & You, 2009] does not hold. Besides, it is proved that the constructive proof of Theorem 4.3 in [Fu & You, 2009] is not correct.

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Coherent local hyperbolicity of a linear extension and the essential spectra of a weighted shift operator on a closed interval

Functional Analysis and Its Applications ◽

10.1007/s10688-005-0013-9 ◽

2005 ◽

Vol 39 (1) ◽

pp. 9-20 ◽

Cited By ~ 5

Author(s):

A. B. Antonevich

Keyword(s):

Linear Extension ◽

Shift Operator ◽

Closed Interval ◽

Weighted Shift ◽

Essential Spectra ◽

Weighted Shift Operator

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The numerical radius of a weighted shift operator

Electronic Journal of Linear Algebra ◽

10.13001/1081-3810.2821 ◽

2015 ◽

Vol 30 ◽

pp. 944-963 ◽

Cited By ~ 1

Author(s):

Batzorig Undrakh ◽

Hiroshi Nakazato ◽

Adiyasuren Vandanjav ◽

Mao-Ting Chien

Keyword(s):

Shift Operator ◽

Point Spectrum ◽

Weighted Shift ◽

Numerical Radius ◽

Weighted Shift Operator ◽

The Real ◽

Weight Sequence

In this paper, the point spectrum of the real Hermitian part of a weighted shift operator with weight sequence a_1, a_2, . . . , a_n, 1, 1, . . . is investigated and the numerical radius of the weighted shift operator in terms of the weighted shift matrix with weights a_1, a_2, . . . , a_n is formulated explicitly.

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On t-entropy and variational principle for the spectral radius of weighted shift operators

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385709000716 ◽

2009 ◽

Vol 30 (5) ◽

pp. 1331-1342 ◽

Cited By ~ 2

Author(s):

V. I. BAKHTIN

Keyword(s):

Dynamical System ◽

Variational Principle ◽

Spectral Radius ◽

Shift Operator ◽

Weighted Shift ◽

Topological Pressure ◽

Weighted Shift Operator ◽

Shift Operators ◽

Dual Functional ◽

Discrete Time Dynamical Systems

AbstractIn this paper we introduce a new functional invariant of discrete time dynamical systems—the so-called t-entropy. The main result is that this t-entropy is the Legendre dual functional to the logarithm of the spectral radius of the weighted shift operator on L1(X,m) generated by the dynamical system. This result is called the variational principle and is similar to the classical variational principle for the topological pressure.

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