Hilbert Space Of Dirichlet Series Research Articles

Compact linear combinations of composition operators on Hilbert spaces of Dirichlet series

Compact linear combinations of composition operators on Hilbert spaces of Dirichlet series

Composition operators on weighted Hilbert spaces of Dirichlet series

AbstractWe study composition operators of characteristic zero on weighted Hilbert spaces of Dirichlet series. For this purpose, we demonstrate the existence of weighted mean counting functions associated with the Dirichlet series symbol, and provide a corresponding change of variables formula for the composition operator. This leads to natural necessary conditions for the boundedness and compactness. For Bergman‐type spaces, we are able to show that the compactness condition is also sufficient, by employing a Schwarz‐type lemma for Dirichlet series.

Algebras and Banach spaces of Dirichlet series with maximal Bohr’s strip

We study linear and algebraic structures in sets of Dirichlet series with maximal Bohr’s strip. More precisely, we consider a set $${\mathscr {M}}$$ of Dirichlet series which are uniformly continuous on the right half plane and whose strip of uniform but not absolute convergence has maximal width, i.e., $$\nicefrac {1}{2}$$ . Considering the uniform norm, we show that $${\mathscr {M}}$$ contains an isometric copy of $$\ell _1$$ (except zero) and is strongly $$\aleph _0$$ -algebrable. Also, there is a dense $$G_\delta $$ set such that any of its elements generates a free algebra contained in $${\mathscr {M}}\cup \{0\}$$ . Furthermore, we investigate $$\mathscr {M}$$ as a subset of the Hilbert space of Dirichlet series whose coefficients are square-summable. In this case, we prove that $${\mathscr {M}}$$ contains an isometric copy of $$\ell _2$$ (except zero).

Orthogonal decomposition of composition operators on the H2 space of Dirichlet series

Let H2 denote the Hilbert space of Dirichlet series with square-summable coefficients. We study composition operators Cφ on H2 which are generated by symbols of the form φ(s)=c0s+∑n≥1cnn−s, in the case that c0≥1. If only a subset P of prime numbers features in the Dirichlet series of φ, then the operator Cφ admits an associated orthogonal decomposition. Under sparseness assumptions on P we use this to asymptotically estimate the approximation numbers of Cφ. Furthermore, in the case that φ is supported on a single prime number, we affirmatively settle the problem of describing the compactness of Cφ in terms of the ordinary Nevanlinna counting function. We give detailed applications of our results to affine symbols and to angle maps.

Linear combinations of composition operators with linear symbols on a Hilbert space of Dirichlet series

On the Hilbert space of Dirichlet series with square summable coefficients, we show that the linear combinations of two composition operators induced by linear symbols are compact only when each one of them is compact. Moreover, such rigid behavior holds partially for some more general symbols.

Dynamics of Composition Operators on Hilbert Spaces of Dirichlet Series

Dynamics of Composition Operators on Hilbert Spaces of Dirichlet Series

Complex symmetry of weighted composition operators on a Hilbert space of Dirichlet series

Complex symmetry of weighted composition operators on a Hilbert space of Dirichlet series

Weighted composition operators on the Hilbert space of Dirichlet series

In this paper, we study weighted composition operators on the Hilbert space of Dirichlet series with square summable coefficients. The Hermitianness, Fredholmness and invertibility of such operators are characterized, and the spectra of compact and invertible weighted composition operators are also described.

Toeplitzness of composition operators on a Hilbert space of Dirichlet series

In this paper, we investigate the -Toeplitzness and asymptotic -Toeplitzness of composition operators and their adjoints on the Hilbert space of Dirichlet series with square summable coefficients. The operator is a shift specific to that space. We obtain characterizations of -Toeplitz and -asymptotically Toeplitz composition operators, where the asymptotic concepts are relative to the uniform, strong, respectively, and weak operator topologies. A characterization of strong asymptotic -Toeplitzness of adjoints of composition operators is also obtained.

Spaces of Dirichlet series with the complete Pick property

We consider reproducing kernel Hilbert spaces of Dirichlet series with kernels of the form $$k\left( {s,u} \right) = \sum {{a_n}} {n^{ - s - \overline u }}$$ , and characterize when such a space is a complete Pick space. We then discuss what it means for two reproducing kernel Hilbert spaces to be “the same”, and introduce a notion of weak isomorphism. Many of the spaces we consider turn out to be weakly isomorphic as reproducing kernel Hilbert spaces to the Drury–Arveson space H 2 in d variables, where d can be any number in {1, 2,...,∞}, and in particular their multiplier algebras are unitarily equivalent to the multiplier algebra of H 2 . Thus, a family of multiplier algebras of Dirichlet series is exhibited with the property that every complete Pick algebra is a quotient of each member of this family. Finally, we determine precisely when such a space of Dirichlet series is weakly isomorphic as a reproducing kernel Hilbert space to H 2 and when its multiplier algebra is isometrically isomorphic to Mult(H 2 ).

Complex symmetric composition operators on a Hilbert space of Dirichlet series

On the Hilbert space of Dirichlet series with square summable coefficients, there are no non-normal complex symmetric composition operators induced by non-constant symbols. This is in sharp contrast with the phenomenon on the classical Hardy space over the unit disk.

Topological Structure of the Spaces of Composition Operators on Hilbert Spaces of Dirichlet Series

In this paper we study some topological properties of the space of bounded composition operators on some Hilbert spaces of Dirichlet series. We first obtain formulas for the norms and essential norms of composition operators and differences of composition operators on Hilbert spaces of Dirichlet series. Then we give a characterization of the isolated points in the topological space of bounded composition operators on some Hilbert spaces of Dirichlet series. Finally we obtain sufficient conditions such that two composition operators are in the same path component. We show, among other results, that all compact composition operators are in the same path component. For a certain class of frequencies we give complete description of path components.

Invariant subspaces of composition operators on a Hilbert space of Dirichlet series

In this paper, we study invariant subspaces of composition operators on the Hilbert space of Dirichlet series with square summable coefficients. The structure of invariant subspaces of a composition operator is characterized, and the strongly closed algebras generated by some composition operators with irrational symbols are shown to be reflexive. As an application, we provide a criterion for composition operators with certain symbols not to be algebraic.

Some properties of composition operators on Hilbert spaces of Dirichlet series

In this paper, we study some properties of composition operators on Hilbert spaces of Dirichlet series, which include the Fredholmness, Hilbert-Schmidtness, spectra, cyclic and hypercyclic phenomenons, and also answer a norm question raised by Cowen and MacCluer.

Zeros of functions in Hilbert spaces of Dirichlet series

The Dirichlet–Hardy space \({\fancyscript{H}}^2\) consists of those Dirichlet series \(\sum _n a_n n^{-s}\) for which \(\sum _n |a_n|^2 1/2\) is a necessary and sufficient condition for the existence of a nontrivial function \(f\) in \({\fancyscript{H}}^2\) vanishing on a given bounded sequence. The proof implies in fact a stronger result: every function in the Hardy space \(H^2\) of the half-plane \(\text{ Re} s>1/2\) can be interpolated by a function in \({\fancyscript{H}}^2\) on such a Blaschke sequence. Analogous results are proved for the Hilbert space \({\fancyscript{D}}_\alpha \) of Dirichlet series \(\sum _n a_n n^{-s}\) for which \(\sum _n |a_n|^2[d(n)]^\alpha 1/2\). Partial results are then obtained for the zeros of functions in \({\fancyscript{H}}^p\) (\(L^p\) analogues of \({\fancyscript{H}}^2\)) for \(2<p<\infty \), based on certain contractive embeddings of \({\fancyscript{D}}_\alpha \) in \({\fancyscript{H}}^p\).

Local properties of Hilbert spaces of Dirichlet series

We show that the asymptotic behavior of the partial sums of a sequence of positive numbers determine the local behavior of the Hilbert space of Dirichlet series defined using these as weights. This extends results recently obtained describing the local behavior of Dirichlet series with square summable coefficients in terms of local integrability, boundary behavior, Carleson measures and interpolating sequences. As these spaces can be identified with functions spaces on the infinite-dimensional polydisk, this gives new results on the Dirichlet and Bergman spaces on the infinite-dimensional polydisk, as well as the scale of Besov–Sobolev spaces containing the Drury–Arveson space on the infinite-dimensional unit ball. We use both techniques from the theory of sampling in Paley–Wiener spaces, and classical results from analytic number theory.

Local interpolation in Hilbert spaces of Dirichlet series

We denote by H \mathscr {H} the Hilbert space of ordinary Dirichlet series with square-summable coefficients. The main result is that a bounded sequence of points in the half-plane σ &gt; 1 / 2 \sigma &gt;1/2 is an interpolating sequence for H \mathscr {H} if and only if it is an interpolating sequence for the Hardy space H 2 H^2 of the same half-plane. Similar local results are obtained for Hilbert spaces of ordinary Dirichlet series that relate to Bergman and Dirichlet spaces of the half-plane σ &gt; 1 / 2 \sigma &gt;1/2 .

Compactness of composition operators on a Hilbert space of Dirichlet series

In this paper new L α p → L β q estimates are proved for translation-invariant Radon transforms along curves for α⩽ β and p< q. For a fixed α and β, if p is sufficiently close to 2 the best possible q is obtained, up to ε. The method is related to that of [5].

Numerical range of composition operators on a Hilbert space of Dirichlet series

We study the numerical range of composition operators on a Hilbert space of Dirichlet series with square-summable coefficients. We first describe the numerical range of “nice” composition operators (as invertible, normal and isometric ones). We also focus on the zero-inclusion question for more general symbols.

Hilbert spaces of Dirichlet series and their multipliers

We consider various Hilbert spaces of Dirichlet series whose norms are given by weighted ℓ 2 \ell ^2 norms of the Dirichlet coefficients. We describe the multiplier algebras of these spaces. The functions in the multiplier algebra may or may not extend to be analytic on a larger half-plane than the functions in the Hilbert space.