instituto de matemáticas universidad de sevilla
Antonio de Castro Brzezicki
Hardy spaces of Dirichlet series and their operators: a survey
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 $\varphi\in L^{2}(0,1)=:H$ be an odd,  $2$-periodic, function. When do its dilates   $\varphi_n,  \varphi_{n}(x)=\varphi(nx), \ n=1,2,$ form a Riesz basis in $H$?  The answer involves the Hardy space $\mathcal{H}^2$ of Dirichlet series and its multiplier space  $\mathcal{H}^\infty$. 
Later (1999), Gordon and Hedenmalm characterized the bounded composition operators $C_\varphi, C_{\varphi}(f)=f\circ \varphi$, on  $\mathcal{H}^2$, and shortly afterwards (2002), Bayart defined $\mathcal{H}^p$ 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  $\mathcal{H}^p$ 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 $1\leq p<2$, and a full description of composition operators on $\mathcal{H}^p, 1\leq p<2$, remains to be given.  In this survey, we shall discuss some aspects of those questions.