DILATION OF RITT OPERATORS ON L-P-SPACES

作者：Arhancet Cedric^*; Le Merdy Christian

来源：Israel Journal of Mathematics, 2014, 201(1): 373-414.

DOI：10.1007/s11856-014-1030-6

For any Ritt operator T: L-P (Omega) -> L-P (Omega), for any positive real number a, and for any x is an element of L-P (Omega), we consider
vertical bar vertical bar x vertical bar vertical bar T,alpha = vertical bar vertical bar((infinity)Sigma(k2 alpha-1)(k=1)vertical bar Tk-1(I-T)(alpha)x vertical bar(2))(1/2)vertical bar vertical bar(Lp)
We show that if T is actually an R-Ritt operator, then the square functions vertical bar vertical bar vertical bar vertical bar(T,alpha),. are pairwise equivalent. Then we show that T and its adjoint T*: L-P' (Omega) -> L-P' (Omega) both satisfy uniform estimates vertical bar x vertical bar vertical bar T,1 less than or similar to vertical bar x vertical bar L-p and vertical bar vertical bar y vertical bar vertical bar T*, 1 less than or similar to vertical bar vertical bar y vertical bar vertical bar L-P 'for x is an element of L-P (Omega) and y is an element of L-P' (Omega) if and only if T is R-Ritt and admits a dilation in the following sense: there exist a measure space 6, an isomorphism U: L-P ((Omega) over tilde) > L-P ((Omega) over tilde) such that {U-n : n E Z} is bounded, as well as two bounded maps L-P (Omega) (J)-> LP ((Omega) over tilde) (Q)-> L-P (Omega) such that T-n = QU(n) J for any n >= 0. We also investigate functional calculus properties of Ritt operators and analogs of the above results on noncommutative L-P-spaces.

• 出版日期2014-1