Mean ergodic operators and reflexive Fréchet lattices

Abstract

Connections between (positive) mean ergodic operators acting in Banach lattices and properties of the underlying lattice itself are well understood; see the works of Emel'yanov, Wol¤ and Zaharopol cited in the references. For Fréchet lattices (or more general locally convex solid Riesz spaces) there is virtually no information available. For a Fréchet lattice E, it is shown here (amongst other things) that every power bounded linear operator on E is mean ergodic if and only if E is re ‡exive if and only if E is Dedekind -complete and every positive power bounded operator on E is mean ergodic if and only if every positive power bounded operator in the strong dual E 0 (no longer a Fréchet lattice) is mean ergodic. An important technique is to develop criteria which detect when E admits a (positively) complemented lattice copy of c0,`1 or`1.

Mean ergodic composition operators in spaces of homogeneous polynomials

“…Fix 1 ≤ p < ∞ and take 0 < α < 1/p. The unilateral weighted backward shift is the operator ϕ α : ℓ p → ℓ p defined by (6) ϕ α (e 1 ) = 0 and ϕ α (e k ) = k k − 1…”

Linear Operators on the (LB)-Sequence Spaces $$\mathbf{ces(p-), 1< p \le \infty } $$

“…The spaces p− and ces( p−), for 1 < p ≤ ∞, also have other desirable Riesz space properties. For instance, they cannot contain an isomorphic lattice copy of either ∞ , 1 or c 0 , [17,Theorem 1.2]. Equivalently, they cannot contain any positively complemented lattice copy of ∞ , 1 , c 0 , [17, Remark 2.5(i) and Proposition 3.2].…”

“…Fix 1 < p ≤ ∞. A consequence of the fact that not every topologically bounded, disjoint sequence in p− converges to 0 is that there must exist a power bounded operator T belonging to the centre Z ( p− ) ⊆ L ( p− ) of p− which is not uniformly mean ergodic, [17,Theorem 5.1]. Recall, T ∈ Z ( p− ) means that there exists 0 ≤ λ ∈ R such that |T (x)| ≤ λ|x| for all x ∈ p− , [17, p. 914].…”

Characterizing Fréchet–Schwartz spaces via power bounded operators

“…A continuous linear operator T acting in a Fréchet space X is called power bounded ( The result of Emel'yanov was extended in [10], where it was shown that a Fréchet lattice X is reexive if and only if every power bounded operator on X is mean ergodic. An analogue of (i) is also presented in [10]. Namely, a discrete Fréchet lattice X is Montel (i.e., bounded sets are relatively compact) if and only if every power bounded operator lying in the centre Z(X) of X is uniformly mean ergodic.…”