On the (C,α) uniform ergodic theorem

Abstract

We improve a recent result of T. Yoshimoto about the uniform ergodic theorem with Cesàro means of order α. We give a necessary and sufficient condition for the (C, α) uniform ergodicity with α > 0. Introduction. In his classical paper [D], N. Dunford obtained several theorems about convergence of (f n (T)) n∈N , where T is a bounded linear operator on a Banach space and (f n) n∈N is a sequence of complex-valued functions, each of which is holomorphic on some open neighborhood of σ(T). Different kinds of convergence (namely, convergence in B(X), strong and weak convergence) were treated. In connection with this, E. Hille [H] obtained, as an application of Abelian and Tauberian theorems, the uniform ergodic theorem as stated below with a view to relating the (C, α) ergodic theorem for an operator T and the properties of the resolvent R(•, T).

Uniformly ergodic A-contractions on Hilbert spaces

“…Similarly, if the limit (1.1) is uniform on the unit ball and Q is an orthogonal projection, we say that T is orthogonally uniformly ergodic on H. This is a stronger concept than the one studied by N. Dunford [3], E. Ed-Dari [4], S. Grabiner and J. Zemánek [6], M. Lin [8], M. Mbekhta and J. Zemánek [9], Y. Tomilov and J. Zemánek [15], J. Zemánek [16] and other authors, in the context of Banach spaces. (1.4) where…”

Operator inequalities I. Models and ergodicity

“…it is not power bounded (see [18,Section 4.7]). Properties, characterization through functional calculus and ergodic results for (C, a)bounded operators can be found in [3,5,13,15,16,18,22] and references therein. Definition 6.5.…”

Growth orders and ergodicity for absolutely Cesàro bounded operators

“…We denote by T the discrete semigroup given by the natural powers of the operator T, that is, T (n) := T n for n ∈ N 0 . Recall ( [1,2,9,10,15,22]) that the Cesàro sum of order α ≥ 0 of T is the family of operators (∆ −α T (n)) n∈N 0 ⊂ B(X) given by ∆ −α T (n)x := (k α * T )(n)x = n j=0 k α (n − j)T j x, x ∈ X, n ∈ N 0 , and the Cesàro mean of order α ≥ 0 of T is the family of operators (M α T (n)) n∈N 0 given by…”

“…In the case that (M α T (n)) n∈N 0 converges in the strong topology of X, we say that the operator T is (C, α)-mean ergodic (for α = 1 it said to be mean ergodic). Properties, characterization thorough functional calculus and ergodic results for (C, α)-Cesàro bounded operators can be found in [2,3,7,8,9,10,15] and references therein. Also, there are connections between some boundedness of the Cesàro means and the resolvent of T. We recall that the condition…”