2011/07/18 by Cédric Arhancet, Arhancet, Cédric · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Operator Algebras (math.OA) #math.FA #math.OA
paper · pdf · doi:10.48550/arxiv.1107.3415
minor corrections; 22 pages; to appear in Mathematica Scandinavica. arXiv admin note: text overlap with arXiv:math/0601645 by other authors
openalex publication_date 2011/07/18 · arxiv created 2012/02/18 · arxiv updated 2012/02/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For any Ritt operator T acting on a noncommutative Lp-space, we define the notion of completely bounded functional calculus H^∞(Bγ) where Bγ is a Stolz domain. Moreover, we introduce the `column square functions' \normxT,c,α=\Bnorm(∑k=1+∞k2α-1|Tk-1(I-T)α(x)|2)1/2Lp(M) and the `row square functions' \normxT,r,α=\Bnorm(∑k=1+∞k2α-1 |(Tk-1(I-T)α(x))^*|2)1/2Lp(M) for any α>0 and any x∈ Lp(M). Then, we provide an example of Ritt operator which admits a completely bounded H^∞(Bγ) functional calculus for some γ∈ ]0,\fracπ2[ such that the square functions \norm⋅T,c,α and \norm⋅T,r,α are not equivalent. Moreover, assuming 1<p<2 and α>0, we prove that if \Ran (I-T) is dense and T admits a completely bounded H^∞(Bγ) functional calculus for some γ∈ ]0,\fracπ2[ then there exists a positive constant C such that for any x ∈ Lp(M), there exists x1, x2 ∈ Lp(M) satisfying x=x1+x2 and \normx1T,c,α+\normx2T,r,α≤ C \normxLp(M). Finally, we observe that this result applies to a suitable class of selfadjoint Markov maps on noncommutative Lp-spaces.