Recurrence of Cosine Operator Functions on Groups

AbstractIn this note, we study the recurrence and topologically multiple recurrence of a sequence of operators on Banach spaces. In particular, we give a sufficient and necessary condition for a cosine operator function, induced by a sequence of operators on the Lebesgue space of a locally compact group, to be topologically multiply recurrent.

Download Full-text

Chaos for Cosine Operator Functions Generated by Shifts

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127414501089 ◽

2014 ◽

Vol 24 (09) ◽

pp. 1450108 ◽

Cited By ~ 2

Author(s):

Chung-Chuan Chen

Keyword(s):

Operator Function ◽

Weighted Shift ◽

Necessary Condition ◽

Weighted Shifts ◽

Cosine Operator Function ◽

Cosine Operator ◽

Operator Functions ◽

Cosine Operator Functions ◽

Sufficient And Necessary Condition

Let 1 ≤ p < ∞. We give the sufficient and necessary condition for cosine operator functions, generated by bilateral weighted shifts on ℓp(ℤ), to be chaotic. Moreover, such a cosine operator function is chaotic if, and only if, its weighted shift is chaotic.

Download Full-text

Chaos for Cosine Operator Functions on Groups

Abstract and Applied Analysis ◽

10.1155/2014/603234 ◽

2014 ◽

Vol 2014 ◽

pp. 1-7 ◽

Cited By ~ 1

Author(s):

Chung-Chuan Chen

Keyword(s):

Locally Compact Group ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Locally Compact ◽

Direct Sum ◽

Cosine Operator ◽

Weight Condition ◽

Operator Functions ◽

Cosine Operator Functions ◽

Necessary And Sufficient

Let1≤p<∞andGbe a locally compact group. We characterize chaotic cosine operator functions, generated by weighted translations on the Lebesgue spaceLp(G), in terms of the weight condition. In particular, chaotic cosine operator functions and chaotic weighted translations can only occur simultaneously. We also give a necessary and sufficient condition for the direct sum of a sequence of cosine operator functions to be chaotic.

Download Full-text

On a cosine operator function framework of approximation processes in Banach space

Filomat ◽

10.2298/fil1913213k ◽

2019 ◽

Vol 33 (13) ◽

pp. 4213-4228

Author(s):

Andi Kivinukk ◽

Anna Saksa ◽

Maria Zeltser

Keyword(s):

Banach Space ◽

Operator Function ◽

Order Of Approximation ◽

Linear Combinations ◽

Cosine Operator Function ◽

Cosine Operator ◽

Operator Functions ◽

Cosine Operator Functions ◽

The Given ◽

General Method

We introduce the cosine-type approximation processes in abstract Banach space setting. The historical roots of these processes go back to W. W. Rogosinski in 1926. The given new definitions use a cosine operator functions concept. We proved that in presented setting the cosine-type operators possess the order of approximation, which coincide with results known in trigonometric approximation. Moreover, a general method for factorization of certain linear combinations of cosine operator functions is presented. The given method allows to find the order of approximation using the higher order modulus of continuity. Also applications for the different type of approximations are given.

Download Full-text

Dynamics of weighted translations generated by group actions

Filomat ◽

10.2298/fil2007131c ◽

2020 ◽

Vol 34 (7) ◽

pp. 2131-2139

Author(s):

Chung-Chuan Chen ◽

Seyyed Tabatabaie ◽

Ali Mohammadi

Keyword(s):

Group Actions ◽

Locally Compact Groups ◽

Necessary Condition ◽

Compact Groups ◽

Locally Compact ◽

Measure Spaces ◽

Operator Functions ◽

Cosine Operator Functions ◽

Dynamical Properties ◽

Sufficient And Necessary Condition

In this paper, we consider actions of locally compact groups on measure spaces, and give a sufficient and necessary condition for weighted translations on such spaces to be chaotic. Moreover, some dynamical properties for certain cosine operator functions, generated by translations, are proved as well.

Download Full-text