On solvability of the cohomology equation in function spaces

Abstract

Abstract. Let T be an endomorphism of a probability measure space (Ω, A, µ), and f be a real-valued measurable function on Ω. We consider the cohomology equationConditions for the existence of real-valued measurable solutions h in some function spaces are deduced. The results obtained generalize and improve a recent result of Alonso, Hong and Obaya.

The cohomological equations in nonsmooth categories

“…Theorem 5.6 generalizes Theorem 2 from [14] where the L p -solutions are constructed for 1 < p < ∞ and then an L ∞ -solution is obtained as p → ∞. For other results concerning L ∞ -solutions see [45] and [58].…”

On Ergodic Averages and the Range of a Closed Operator

“…We then choose a positive integer n so that γ ≤ n. By (15) we have C n t ≤ M for all t > 0. Thus, by (14), for λ ∈ C with λ > 0 we can define a bounded linear operator R(λ) on X by ( 17) T sn x ds n ) .…”

“…By (15) Hence {A t } is an A-ergodic net satisfying condition (c), and {B t } is its companion net. It is also easily checked that {B t } satisfies condition (d).…”

“…By using ergodic theory, many authors have studied the problem of solving the functional equation Ax = y for a given y ∈ X. See, for example, Alonso, Hong and Obaya [1], Assani [2], Dotson [4][5][6], Krengel and Lin [10], Lin and Sine [12], Sato [14][15][16][17][18], Shaw [19,20], and Shaw and Li [21]. In particular, Shaw [19] (see also Dotson [4] and Shaw-Li [21]) studied deeply the mean ergodic properties of an A-ergodic net {A α } and its companion net {B α } consisting of bounded linear operators on X, and applied them to the problem successfully.…”