Hardy Spaces Of Dirichlet Series Research Articles

Multipliers for Hardy spaces of Dirichlet series

Multipliers for Hardy spaces of Dirichlet series

Schatten class composition operators on the Hardy space of Dirichlet series and a comparison-type principle

We give necessary and sufficient conditions for a composition operator with Dirichlet series symbol to belong to the Schatten classes S_{p} of the Hardy space \mathcal{H}^{2} of Dirichlet series. For p\geq 2 , these conditions lead to a characterization for the subclass of symbols with bounded imaginary parts. Finally, we establish a comparison-type principle for composition operators. Applying our techniques in conjunction with classical geometric function theory methods, we prove the analogue of the polygonal compactness theorem for \mathcal{H}^{2} and we give examples of bounded composition operators with Dirichlet series symbols on \mathcal{H}^{p} , p>0 .

Counting functions for Dirichlet series and compactness of composition operators

AbstractWe give a sufficient condition for a composition operator with positive characteristic to be compact on the Hardy space of Dirichlet series.

Semigroups of composition operators on Hardy spaces of Dirichlet series

We consider continuous semigroups of analytic functions {Φt}t≥0 in the so-called Gordon-Hedenmalm classG, that is, the family of analytic functions Φ:C+→C+ giving rise to bounded composition operators in the Hardy space of Dirichlet series H2. We show that there is a one-to-one correspondence between continuous semigroups {Φt}t≥0 in the class G and strongly continuous semigroups of composition operators {Tt}t≥0, where Tt(f)=f∘Φt, f∈H2. We extend these results for the range p∈[1,∞). For the case p=∞, we prove that there is no non-trivial strongly continuous semigroup of composition operators in H∞. We characterize the infinitesimal generators of continuous semigroups in the class G as those Dirichlet series sending C+ into its closure. Some dynamical properties of the semigroups are obtained from a description of the Koenigs map of the semigroup.

Composition Operators on Hilbert Spaces of Dirichlet Series

Motivated by a theorem of Gordon and Hedenmalm in 1999, the study of composition operators acting on various scales of function spaces of Dirchlet series has arisen intensive interest. In this paper, we characterize the boundedness of composition operators induced by specific Dirichlet series symbols from Bergman space to Hardy space of Dirichlet series.

A Montel-Type Theorem for Hardy Spaces of Holomorphic Functions

We give a version of the Montel theorem for Hardy spaces of holomorphic functions on an infinite dimensional space. Precisely, we show that any bounded sequence of holomorphic functions in some Hardy space, has a subsequence that converges uniformly over compact subsets to a function that also belongs to the same Hardy space. As a by-product of our results for spaces of functions on infinitely many variables, we also provide an elementary proof of a Montel-type theorem for the Hardy space of Dirichlet series.

Convergence and almost sure properties in Hardy spaces of Dirichlet series

Given a frequency $$\lambda $$ , we study general Dirichlet series $$\sum a_n e^{-\lambda _n s}$$ . First, we give a new condition on $$\lambda $$ which ensures that a somewhere convergent Dirichlet series defining a bounded holomorphic function in the right half-plane converges uniformly in this half-plane, improving classical results of Bohr and Landau. Then, following recent works of Defant and Schoolmann, we investigate Hardy spaces of these Dirichlet series. We get general results on almost sure convergence which have an harmonic analysis flavour. Nevertheless, we also exhibit examples showing that it seems hard to get general results on these spaces as spaces of holomorphic functions.

A mean counting function for Dirichlet series and compact composition operators

We introduce a mean counting function for Dirichlet series, which plays the same role in the function theory of Hardy spaces of Dirichlet series as the Nevanlinna counting function does in the classical theory. The existence of the mean counting function is related to Jessen and Tornehave's resolution of the Lagrange mean motion problem. We use the mean counting function to describe all compact composition operators with Dirichlet series symbols on the Hardy–Hilbert space of Dirichlet series, thus resolving a problem which has been open since the bounded composition operators were described by Gordon and Hedenmalm. The main result is that such a composition operator is compact if and only if the mean counting function of its symbol satisfies a decay condition at the boundary of a half-plane.

Hardy–Orlicz Spaces of Dirichlet Series: An Interpolation Problem on Abscissae of Convergence

Abstract The study of Hardy spaces of Dirichlet series denoted by $\mathscr{H}^p$ ($p\geq 1$) was initiated in [7] when $p=2$ and $p=\infty $, and in [2] for the general case. In this paper we introduce the Orlicz version of spaces of Dirichlet series $\mathscr{H}^\psi $. We focus on the case $\psi =\psi _q(t)=\exp (t^q)-1,$ and we compute the abscissa of convergence for these spaces. It turns out that its value is $\min \{1/q\,,1/2\}$ filling the gap between the case $\mathscr{H}^\infty $, where the abscissa is equal to $0$, and the case $\mathscr{H}^p$ for $p$ finite, where the abscissa is equal to $1/2$. The upper-bound estimate relies on an elementary method that applies to many spaces of Dirichlet series. This answers a question raised by Hedenmalm in [6].

Interpolation of Hardy spaces of Dirichlet series

We initiate the study of interpolation of the Hardy spaces of Dirichlet series. We prove that there are no variants of the classical interpolation theorems in the setting of Hardy spaces on the countably infinite polytorus T∞ as in the case of the Hardy spaces on the finite dimensional torus. As a byproduct of our results we get a similar conclusion for the Hardy spaces of Dirichlet series by using the famous Bohr lift. Our proofs rely on a general method based on interpolation estimates for finite-dimensional Banach spaces. This method is applied to interpolation of injective tensor products and also of spaces of m-homogeneous polynomials.

Fatou and brothers Riesz theorems in the infinite-dimensional polydisc

We study the boundary behavior of functions in the Hardy spaces on the infinite-dimensional polydisc. These spaces are intimately related to the Hardy spaces of Dirichlet series. We exhibit several Fatou and Marcinkiewicz- Zygmund type theorems for radial convergence of functions with Fourier spectrum supported on $$\mathbb{N}_0^\infty\cup(-\mathbb{N}_0^\infty)$$ . As a consequence one obtains easy new proofs of the brothers F. and M. Riesz Theorems in infinite dimensions, as well as being able to extend a result of Rudin concerning which functions are equal to the modulus of an H1 function almost everywhere to $$\mathbb{T}^\infty$$ . Finally, we provide counterexamples showing that the pointwise Fatou theorem is not true in infinite dimensions without restrictions to the mode of radial convergence even for bounded analytic functions.

A note on abscissas of Dirichlet series

We present an abstract approach to the abscissas of convergence of vector-valued Dirichlet series. As a consequence we deduce that the abscissas for Hardy spaces of Dirichlet series are all equal. We also introduce and study weak versions of the abscissas for scalar-valued Dirichlet series.

Contractive inequalities for Bergman spaces and multiplicative Hankel forms

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.

Sharp norm estimates for composition operators and Hilbert-type inequalities

Let $\mathscr{H}^2$ denote the Hardy space of Dirichlet series $f(s) = \sum_{n\geq1} a_n n^{-s}$ with square summable coefficients and suppose that $\varphi$ is a symbol generating a composition operator on $\mathscr{H}^2$ by $\mathscr{C}_\varphi(f) = f \circ \varphi$. Let $\zeta$ denote the Riemann zeta function and $\alpha_0=1.48\ldots$ the unique positive solution of the equation $\alpha\zeta(1+\alpha)=2$. We obtain sharp upper bounds for the norm of $\mathscr{C}_\varphi$ on $\mathscr{H}^2$ when $0<\operatorname{Re}\varphi(+\infty)-1/2 \leq \alpha_0$, by relating such sharp upper bounds to the best constant in a family of discrete Hilbert-type inequalities.

Compact composition operators with nonlinear symbols on the H2 space of Dirichlet series

We investigate the compactness of composition operators on the Hardy space of Dirichlet series induced by a map $\varphi(s)=c_0s+\varphi_0(s)$, where $\varphi_0$ is a Dirichlet polynomial. Our results depend heavily on the characteristic $c_0$ of $\varphi$ and, when $c_0=0$, on both the degree of $\varphi_0$ and its local behaviour near a boundary point. We also study the approximation numbers for some of these operators. Our methods involve geometric estimates of Carleson measures and tools from differential geometry.

Multipliers of Dirichlet series and monomial series expansions of holomorphic functions in infinitely many variables

Let $\mathcal{H}_\infty$ be the set of all ordinary Dirichlet series $D=\sum_n a_n n^{-s}$ representing bounded holomorphic functions on the right half plane. A multiplicative sequence $(b_n)$ of complex numbers is said to be an $\ell_1$-multiplier for $\mathcal{H}_\infty$ whenever $\sum_n |a_n b_n| < \infty$ for every $D \in \mathcal{H}_\infty$. We study the problem of describing such sequences $(b_n)$ in terms of the asymptotic decay of the subsequence $(b_{p_j})$, where $p_j$ denotes the $j$th prime number. Given a multiplicative sequence $b=(b_n)$ we prove (among other results): $b$ is an $\ell_1$-multiplier for $\mathcal{H}_\infty$ provided $|b_{p_j}| < 1$ for all $j$ and $\overline{\lim}_n \frac{1}{\log n} \sum_{j=1}^n b_{p_j}^{*2} < 1$, and conversely, if $b$ is an $\ell_1$-multiplier for $\mathcal{H}_\infty$, then $|b_{p_j}| < 1$ for all $j$ and $\overline{\lim}_n \frac{1}{\log n} \sum_{j=1}^n b_{p_j}^{*2} \leq 1$ (here $b^*$ stands for the decreasing rearrangement of $b$). Following an ingenious idea of Harald Bohr it turns out that this problem is intimately related with the question of characterizing those sequences $z$ in the infinite dimensional polydisk $\mathbb{D}^\infty$ (the open unit ball of $\ell_\infty$) for which every bounded and holomorphic function $f$ on $\mathbb{D}^\infty$ has an absolutely convergent monomial series expansion $\sum_{\alpha} \frac{\partial_\alpha f(0)}{\alpha!} z^\alpha$. Moreover, we study analogous problems in Hardy spaces of Dirichlet series and Hardy spaces of functions on the infinite dimensional polytorus $\mathbb{T}^\infty$.

Volterra operators on Hardy spaces of Dirichlet series

Abstract For a Dirichlet series symbol g ⁢ ( s ) = ∑ n ≥ 1 b n ⁢ n - s {g(s)=\sum_{n\geq 1}b_{n}n^{-s}} , the associated Volterra operator 𝐓 g {\mathbf{T}_{g}} acting on a Dirichlet series f ⁢ ( s ) = ∑ n ≥ 1 a n ⁢ n - s {f(s)=\sum_{n\geq 1}a_{n}n^{-s}} is defined by the integral f ↦ - ∫ s + ∞ f ⁢ ( w ) ⁢ g ′ ⁢ ( w ) ⁢ 𝑑 w . {f\mapsto-\int_{s}^{+\infty}f(w)g^{\prime}(w)\,dw}. We show that 𝐓 g {\mathbf{T}_{g}} is a bounded operator on the Hardy space ℋ p {\mathcal{H}^{p}} of Dirichlet series with 0 &lt; p &lt; ∞ {0&lt;p&lt;\infty} if and only if the symbol g satisfies a Carleson measure condition. When appropriately restricted to one complex variable, our condition coincides with the standard Carleson measure characterization of BMOA ⁡ ( 𝔻 ) {{\operatorname{BMOA}}(\mathbb{D})} . A further analogy with classical BMO {{\operatorname{BMO}}} is that exp ⁡ ( c ⁢ | g | ) {\exp(c|g|)} is integrable (on the infinite polytorus) for some c &gt; 0 {c&gt;0} whenever 𝐓 g {\mathbf{T}_{g}} is bounded. In particular, such g belong to ℋ p {\mathcal{H}^{p}} for every p &lt; ∞ {p&lt;\infty} . We relate the boundedness of 𝐓 g {\mathbf{T}_{g}} to several other BMO {{\operatorname{BMO}}} -type spaces: BMOA {{\operatorname{BMOA}}} in half-planes, the dual of ℋ 1 {\mathcal{H}^{1}} , and the space of symbols of bounded Hankel forms. Moreover, we study symbols whose coefficients enjoy a multiplicative structure and obtain coefficient estimates for m-homogeneous symbols as well as for general symbols. Finally, we consider the action of 𝐓 g {\mathbf{T}_{g}} on reproducing kernels for appropriate sequences of subspaces of ℋ 2 {\mathcal{H}^{2}} . Our proofs employ function and operator theoretic techniques in one and several variables; a variety of number theoretic arguments are used throughout the paper in our study of special classes of symbols g.

Zeros of Functions in Bergman-Type Hilbert Spaces of Dirichlet Series

For a real number $\alpha$ the Hilbert space $\mathscr{D}_\alpha$ consists of those Dirichlet series $\sum_{n=1}^\infty a_n/n^s$ for which $\sum_{n=1}^\infty |a_n|^2/[d(n)]^\alpha &lt; \infty$, where $d(n)$ denotes the number of divisors of $n$. We extend a theorem of Seip on the bounded zero sequences of functions in $\mathscr{D}_\alpha$ to the case $\alpha&gt;0$. Generalizations to other weighted spaces of Dirichlet series are also discussed, as are partial results on the zeros of functions in the Hardy spaces of Dirichlet series $\mathscr{H}^p$, for $1\leq p &lt;2$.

An embedding constant for the Hardy space of Dirichlet series

A new and simple proof of the embedding of the Hardy–Hilbert space of Dirichlet series into a conformally invariant Hardy space of the half-plane is presented, and the optimal constant of the embedding is computed.

Some Banach spaces of Dirichlet series

The Hardy spaces of Dirichlet series denoted by ${\cal H}^p$ ($p\ge1$) have been studied in [12] when p = 2 and in [3] for the general case. In this paper we study some Lp-generalizations of spaces of Dirichlet series, particularly two families of Bergman spaces denoted ${\cal A}^p$ and ${\cal B}^p$. We recover classical properties of spaces of analytic functions: boundedness of point evaluation, embeddings between these spaces and Littlewood-Paley formulas when p = 2. We also show that the ${\cal B}^p$ spaces have properties similar to the classical Bergman spaces of the unit disk while the ${\cal A}^p$ spaces have a different behavior.