Westudyseveralnotionsofboundedness foroperators. It isknownthat anypowerboundedoperatorisabsolutelyCes`aroboundedandstronglyKreiss bounded(inparticular,uniformlyKreissbounded).Theconversesdonothold ingeneral. Inthisnote,wegiveexamplesoftopologicallymixing(hence,not powerbounded)absolutelyCes`aroboundedoperatorson p(N),1≤p<∞, andprovideexamplesofuniformlyKreissboundedoperatorswhicharenot absolutelyCes`arobounded.Theseresultscomplementafewknownexamples (see [27] and [2]). Wealsoobtainacharacterizationof powerboundedoperatorswhichgeneralizesaresultofVanCasteren [32]. In [2]Alemanand SuciuaskedifeveryuniformlyKreissboundedoperatorTonaBanachspace satisfiesthat limn→∞ Tn n =0.WesolvethisquestionforHilbertspaceoperatorsand,moreover,weprovethat, ifT isabsolutelyCes`aroboundedona Banach(Hilbert)space,then Tn =o(n)( Tn =o(n1 2),respectively).Asa consequence,everyabsolutelyCes`aroboundedoperatoronareflexiveBanach spaceismeanergodic.
