Espaces de séries de Dirichlet et leurs opérateurs de composition

Bull. London Math. Soc.

2017

“…Proof A computation (or [, Theorem 2.31]) shows that if

g \in {scriptH}^{2}

converges uniformly in

C_{0}

, then

{∥ g ∥}_{H^{2}} = \underset{β \to \infty}{trueprefixlim} {∥ g ∥}_{H_{prefixi}^{2} false(C_{0}, β false)} .

In particular, if

f

is a Dirichlet polynomial and

φ

is in

scriptG

with

c_{0} = 0

, then by we get that

false∥ {scriptC}_{φ} {f ∥}_{H^{2}} = \underset{β \to \infty}{trueprefixlim} {∥ f \circ φ ∥}_{H_{prefixi}^{2} false(C_{0}, β false)} = \underset{β \to \infty}{trueprefixlim} {∥ f \circ φ \circ {scriptT}_{β} ∥}_{H^{2} false(double-struckD false)} .

Define

F \in H^{2} false(double-struckD false)

and

ϕ : D \to D

\begin{matrix} true \\ right F & left : = f \circ S_{1 / 2} \circ T_{α} \\ right ϕ & left : = T_{α}^{- 1} \circ S_{1 / 2}^{- 1} \circ φ \circ... \end{matrix}

…”

Section: Proof Of Theoremmentioning

confidence: 93%

“…The study of composition operators on

H^{2}

was initiated by Gordon and Hedenmalm in their pioneering paper (see also ), where they proved that an analytic function

φ : {double-struckC}_{1 / 2} \to {double-struckC}_{1 / 2}

generates a composition operator on

H^{2}

if and only if it is a member of the following class.…”

Section: Introductionmentioning

confidence: 99%

Sharp norm estimates for composition operators and Hilbert-type inequalities

Bull. London Math. Soc.

2017

“…For 1 ≤ p < ∞, we follow [3] and define the Hardy space H p as the Banach space completion of Dirichlet polynomials P (s) = N n=1 a n n −s in the Besicovitch norm The spaces H p are Dirichlet series analogues of the classical Hardy spaces in unit disc. We refer to [17] and to [18,Ch. 6] for basic properties of H p , mentioning for the moment only that their elements are absolutely convergent in the half-plane C 1/2 , where C θ := {s ∈ C : Re(s) > θ}.…”

Section: Introductionmentioning

confidence: 99%

Composition operators and embedding theorems for some function spaces of Dirichlet series

Bayart

2019

Math. Z.

We observe that local embedding problems for certain Hardy and Bergman spaces of Dirichlet series are equivalent to boundedness of a class of composition operators. Following this, we perform a careful study of such composition operators generated by polynomial symbols ϕ on a scale of Bergman-type Hilbert spaces D α . We investigate the optimal β such that the composition operator C ϕ maps D α boundedly into D β . We also prove a new embedding theorem for the non-Hilbertian Hardy space H p into a Bergman space in the half-plane and use it to consider composition operators generated by polynomial symbols on H p , finding the first nontrivial results of this type. The embedding also yields a new result for the functional associated to the multiplicative Hilbert matrix. 1 2 < ∞.Here d (n) denotes the number of divisors of the positive integer n. Note that D 0 = H 2 . We are interested in the range α ≥ 0 and, as explained in [1], these spaces may be thought of as Dirichlet series analogues of the classical scale of weighted Bergman spaces in the unit disc. Since d (n) = O (n ε ) for every ε > 0, it follows from the Cauchy-Schwarz inequality that Dirichlet series in D α also are absolutely convergent in C 1/2 .Due to an insight of H. Bohr (see Section 2), both H p and D α can be identified with certain function spaces in countably infinite number of complex variables, and -consequently -the

“…Hardy spaces of Dirichlet series, H p , are defined by requiring this identification to induce an isometric, multiplicative isomorphism. The connection to Dirichlet series gives rise to a rich interplay between operator theory and analytic number theorywe refer the interested reader to the survey [37] or the monograph [38] as a starting point.…”

Section: Introductionmentioning

confidence: 99%

Contractive inequalities for Bergman spaces and multiplicative Hankel forms

Bayart

Trans. Amer. Math. Soc.

Haimi

et al. 2018

We consider sharp inequalities for Bergman spaces of the unit disc, establishing analogues of the inequality in Carleman's proof of the isoperimetric inequality and of Weissler's inequality for dilations. By contractivity and a standard tensorization procedure, the unit disc inequalities yield corresponding inequalities for the Bergman spaces of Dirichlet series. We use these results to study weighted multiplicative Hankel forms associated with the Bergman spaces of Dirichlet series, reproducing most of the known results on multiplicative Hankel forms associated with the Hardy spaces of Dirichlet series. In addition, we find a direct relationship between the two type of forms which does not exist in lower dimensions. Finally, we produce some counter-examples concerning Carleson measures on the infinite polydisc.