Operator ergodic theory for one-parameter decomposable groups

✍ Scribed by Earl Berkson

• Publisher: Elsevier Science
• Year: 2011
• Tongue: French
• Weight: 239 KB
• Volume: 135
• Category: Article
• ISSN: 0007-4497
• DOI: 10.1016/j.bulsci.2011.03.004

✦ Synopsis

After initial treatment of the Fourier analysis and operator ergodic theory of strongly continuous decomposable one-parameter groups of operators in the Banach space setting, we show that in the setting of a super-reflexive Banach space X these groups automatically transfer from the setting of R to X the behavior of the Hilbert kernel, as well as the Fourier multiplier actions of functions of higher variation on R. These considerations furnish one-parameter groups with counterparts for the single operator theory in Berkson (2010) [4]. Since no uniform boundedness of one-parameter groups of operators is generally assumed in the present article, its results for the super-reflexive space setting go well beyond the theory of uniformly bounded one-parameter groups on UMD spaces (which was developed in Berkson et al., 1986 [13]), and in the process they expand the scope of vector-valued transference to encompass a genre of representations of R that are not uniformly bounded.