Decay of correlations at a speed given by some numbers for some dynamics and two spaces of functions over with zero average, is formulated in two similar ways in various works: for all ,
- for some depending arbitrarily on ;
- for some independent of ;
where is the correlation at time , a bilinear form .
The second type seems stronger than the first (and it is strictly so for arbitrary families of bilinear forms). However the above two types are really equivalent under very general assumptions, i.e., if (i) are Banach spaces and (ii) each is continuous for any given .
Indeed, fix .Let be defined by . Each such map is linear and continuous by (ii). Now, for each , . Using (i), the Banach-Steinhaus theorem gives such that . In other words, , defined by is linear and continuous.
Similarly defined by is linear and continuous. Now, for each , where is the unit ball in . Using (i), the Banach-Steinhaus theorem gives such that for all . But this implies the first type of decay above. CQFD.
This was pointed out to me by Sébastien GOUËZEL.
Jean-René CHAZOTTES points out that this is essentially Theorem B.1 of his paper joint with P. Collet and B. Schmitt: Statistical consequences of the Devroye inequality for processes. Applications to a class of non-uniformly hyperbolic dynamical systems. Nonlinearity 18 (2005).