Dynamics of weighted translations generated by group actions

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.

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

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.

Disjoint topological transitivity for cosine operator functions on groups

In this paper, we study finite sequences of operators, generated by the powers of weighted translations on discrete groups, and give sufficient conditions for such sequences to be disjoint topologically transitive and mixing in terms of the group elements and weights. The sequences of operators are cosine operator functions. Moreover, we also obtain necessary conditions for cosine operator functions to be disjoint transitive and mixing.

Recurrence of Cosine Operator Functions on Groups

In 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.

Chaos for Cosine Operator Functions Generated by Shifts

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.

Chaos for Cosine Operator Functions on Groups

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.