2025/04/01 by Austin Anderson, Mirjana Jovovic, Mirjana Jovović +1
Mathematics · #Mathematical Dynamics and Fractals #Holomorphic and Operator Theory #Advanced Banach Space Theory
paper · pdf · doi:10.1007/s11785-025-01686-7
We study semigroups of composition operators acting on the Besov spaces Bp, where they exhibit some new behaviors relative to many classical spaces. Often for a Banach space X of analytic functions on the unit disk, the maximal closed space of strong continuity, [ φt, X ], exists for every semigroup \ φt \ of analytic self-maps of the disk, and the question whether [φt , X ] equals X itself has an answer independent of \φt\. Such is the case for the Hardy and Bergman spaces, Bloch, BMOA, and H∞. For the disk algebra A, [φt , A ] = A precisely when \φt\ ⊂ A. For Bp with p ≥ 2, every \φt\ ⊂ Bp and always [ φt, Bp ] = Bp, but this fails when 1 < p < 2. We give an example where \φt\ ⊂ Bp and yet the induced composition operators \Ct\ are not bounded on Bp and we do not know if [φt,Bp] exists. If it does exist, it cannot be equal to Bp. Under the hypothesis that there is a uniform bound for the operator norms of the \Ct\, 0 ≤ t ≤ 1, we characterize the semigroups \ φt \ such that [ φt, Bp ] = Bp.