Hardy spaces of Dirichlet series and their operators: a survey

Hardy spaces of Dirichlet series and their operators: a survey
Luis Rodríguez Piazza
Seminario I (IMUS), Edificio Celestino Mutis
Hervé Queffélec

The analytic theory of Dirichlet series was initiated by  H.~Bohr around 1910, and pursued among others   by Kahane and his students in the seventies and eighties. It became somehow dozing in the mid-nineties, till a result of Hedenmalm, Lindqvist, Seip in 1997, giving a full answer to a question of Beurling: Let φ∈L2(0,1)=:Hφ∈L2(0,1)=:H be an odd,  22-periodic, function. When do its dilates   φn,φn(x)=φ(nx), n=1,2,φn,φn(x)=φ(nx), n=1,2, form a Riesz basis in HH?  The answer involves the Hardy space H2H2 of Dirichlet series and its multiplier space  H∞H∞.

Later (1999), Gordon and Hedenmalm characterized the bounded composition operators Cφ,Cφ(f)=f∘φCφ,Cφ(f)=f∘φ, on  H2H2, and shortly afterwards (2002), Bayart defined HpHp spaces, and gave a partial description of their composition operators.\\  The approximation numbers of those operators were also studied by himself, the author and K.~Seip. The multiplicative Hankel operators were independently studied by  Helson till 2010, and then by Pushnitskii, Seip, Vukotic, and al.
The introduction of  HpHp spaces raises some delicate problems, all of which are not solved.  % L'introduction de ces espaces, et l'\'etude de leurs op\'erateurs de composition (Aleman, Olsen, Saksman, Bayart, Queff\'elec, Seip) n\'ecessite un peu de th\'eorie ergodique, de \textit{l'Analyse harmonique} sur le tore de dimension infinie, et soul\`eve des probl\`emes d\'elicats, qui ne sont pas encore tous r\'esolus. Recent progress (positive answer by Harper to  Helson's conjecture in 2017) has been made. As a consequence,  one has a negative  solution to the local embedding conjecture for 1≤p<21≤p<2, and a full description of composition operators on Hp,1≤p<2Hp,1≤p<2, remains to be given.  In this survey, we shall discuss some aspects of those questions.