Square Functions for Ritt Operators on Noncommutative L^p-Spaces
For any Ritt operator T acting on a noncommutative L^p-space, we define the notion of completely bounded functional calculus H^∞(B_γ) where B_γ is a Stolz domain. Moreover, we introduce the ’column square functions’ \[ \|x\|_p,T,c,α=\|(_k=1^+∞k^2α-1 |T^k-1(I-T)^α(x)|^2)^12 \|_L^p(M) \] and the ’row square functions’ \[ \|x\|_p,T,r,α=\|(_k=1^+∞k^2α-1 |(T^k-1(I-T)^α(x))^*|^2)^12 \|_L^p(M) \] for any α>0 and any xL^p(M). Then, we provide an example of Ritt operator which admits a completely bounded H^∞(B_γ) functional calculus for some γ ]0,π2[ such that the square functions \|·\|_p,T,c,α (or \|·\|_p,T,r,α) are not equivalent to the usual norm \|·\|_L^p(M). Moreover, assuming 1<p<2 and α>0, we prove that if Ran olimits (I-T) is dense and T admits a completely bounded H^∞(B_γ) functional calculus for some γ ]0,π2[ then there exists a positive constant C such that for any x L^p(M), there exists x_1, x_2 L^p(M) satisfying x=x_1+x_2 and \|x_1\|_p,T,c,α+\|x_2\|_p,T,r,αC \|x\|_L^p(M). Finally, we observe that this result applies to a suitable class of selfadjoint Markov maps on noncommutative L^p-spaces. , number=2, journal=Mathematica Scandinavica, author=Arhancet, Cédric, year=2013, month=Dec., pages=292–319