Ergodicity and stability of orbits of unbounded semigroup representations

Abstract

We develop a theory of ergodicity for unbounded functions <t> : J -*• X, where J is a subsemigroup of a locally compact abelian group G and X is a Banach space. It is assumed that <j> is continuous and dominated by a weight w denned on G. In particular, we establish total ergodicity for the orbits of an (unbounded) strongly continuous representation T : G ->• L(X) whose dual representation has no unitary point spectrum. Under additional conditions stability of the orbits follows. To study spectra of functions, we use Beurling algebras L^ (G) and obtain new characterizations of their maximal primary ideals, when w is non-quasianalytic, and of their minimal primary ideals, when w has polynomial growth. It follows that, relative to certain translation invariant function classes &', the reduced Beurling spectrum of <j> is empty if and only if # € &. For the zero class, this is Wiener's tauberian theorem.2000 Mathematics subject classification: primary 46J20,43A60; secondary 47A35, 34K25, 28B05.

A spectral countability condition for almost automorphy of solutions of differential equations

“…Suppose that sp A (x) is not empty. SinceS| Mx is an isometry in Mx and sp A (x) = σ(S| Mx ) is countable, by the Gelfand Theorem there is a point z 0 in its spectrum that is an eigenvalue ofS| Mx (for the Gelfand Theorem, see, e.g., [1,3]). So, we can find an elementỹ ∈ Mx such that z 0ỹ =Sỹ.…”

A Spectral Theory of Polynomially Bounded Sequences and Applications to the Asymptotic Behavior of Discrete Systems

“…(1.1) have been obtained, see e.g. [1,3,4,5,6,7,9,11,14,15,16]. Among many interesting results in this direction is a famous theorem due to Katznelson-Tzafriri (see [9]) saying that if T is a bounded operator in a Banach space X such that sup n∈N T n < ∞, (1.…”

On the asymptotic behaviour of Volterra difference equations

“…We refer the reader to the books [3,8], and papers [1,6,7,9,10,11,12,13,15,17,18] and their references for more information in this direction. Related results for differential equations can be found in [2,4,5,14,16].…”