C0-semigroups and mean ergodic operators in a class of Fréchet spaces

Abstract

It is shown that the generator of every exponentially equicontinuous, uniformly continuous C 0 -semigroup of operators in the class of quojection Fréchet spaces X (which includes properly all countable products of Banach spaces) is necessarily everywhere defined and continuous. If, in addition, X is a Grothendieck space with the Dunford-Pettis property, then uniform continuity can be relaxed to strong continuity. Two results, one of M. Lin and one of H.P. Lotz, both concerned with uniformly mean ergodic operators in Banach spaces, are also extended to the class of Fréchet spaces mentioned above. They fail to hold for arbitrary Fréchet spaces.

Distributional Chaoticity of C0-Semigroup on a Frechet Space

“…In Section 4, using upper density and lower density, it is showed that the distributional chaoticity of µ p (T ) = {T t } t≥0 is equivalent to the distributional chaoticity of some T t 0 (t 0 > 0). This result is consistent with the similar conclusion in Banach space or other Frechet spaces (see References [17,27,29,31,33] and others). Then, some further results regarding C 0 -semigroups or Frechet spaces may be obtained in the future.…”

“…Recently, an extension of distributional chaos for a family of operators (including C 0 -semigroups) on Frechet spaces were proposed by Conejero [26]. For other studies of C 0 -semigroups or Frechet spaces see References [27][28][29][30][31][32][33][34] and others.…”