2014/11/03 by N. J. Kalton, Kalton, Nigel, Lutz Weis +1 · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Mathematical functions and polynomials
paper · pdf · doi:10.48550/arxiv.1411.0472
openalex publication_date 2014/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Using notions from the geometry of Banach spaces we introduce square functions γ(Ω,X) for functions with values in an arbitrary Banach space X. We show that they have very convenient function space properties comparable to the Bochner norm of L2(Ω,H) for a Hilbert space H. In particular all bounded operators T on H can be extended to γ(Ω,X) for all Banach spaces X. Our main applications are characterizations of the H∞--calculus that extend known results for Lp--spaces from \citeCowlingDoustMcIntoshYagi. With these square function estimates we show, e. g., that a c0--group of operators Ts on a Banach space with finite cotype has an H∞--calculus on a strip if and only if e-a|s|Ts is R--bounded for some a > 0. Similarly, a sectorial operator A has an H∞--calculus on a sector if and only if A has R--bounded imaginary powers. We also consider vector valued Paley--Littlewood g--functions on UMD--spaces.