G. Belitskii, V. Rayskin 
Equivalence of families of diffeomorphisms on Banach spaces
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ABSTRACT.  We consider two families of $C^\infty$-diffeomorphisms (with 
hyperbolic linear part at $0$) on a Banach space. Suppose that these two families formally conjugate at $0$. We prove that they admit 
local conjugation, which is infinitely smooth in both, the space 
variable and the family parameter. In particular, subject to 
non-resonance condition, there exists a family of local $C^\infty$ 
linearizations of the family of diffeomorphisms. The linearizing 
family has $C^\infty$ smoothness in parameter. The results are proved under the assumption that the Banach space allows a special extension of the maps. We discuss corresponding properties of Banach spaces.