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Register a new userAnalytic symbols associated with the coefficient matrix of a Dirichlet series kernel
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For a matrix $mathbf a=(a_{m, n})_{m, n=1}^{infty},$ the Dirichlet series kernel $kappa_{mathbf a}(s, u)$ is the double Dirichlet series $sum_{m, n =1}^{infty} a_{m, n}m^{-s} n^{-overline{u}}$ in the variables $s$ and $overline{u},$ which is regularly convergent on some right half-plane $mathbb H_{rho}.$ If the coefficient matrix $mathbf a$ of $kappa_{mathbf a}$ is formally positive semi-definite, then there exists a Hilbert space $mathscr H_{mathbf a}$ with the reproducing kernel $kappa_{mathbf a}.$ The analytic symbols $A_{n, mathbf a} = sum_{m=1}^{infty} a_{m, n}m^{-s},$ $n geq 1,$ associated with $mathbf a$ plays a central role in the study of the reproducing kernel Hilbert spaces $mathscr H_{mathbf a}.$ In particular, they form a total subset of $mathscr H_{mathbf a}$ and provide the formula $sum_{n=1}^{infty}langle{f, A_{n, mathbf a}}rangle n^{-s},$ $s in mathbb H_rho,$ for $f$ in $mathscr H_{mathbf a}.$ We also discuss the role of the analytic symbols in the study of Helson matrices generated by a Radon measure on $(0, infty).$ We focus on two families of Helson matrices; one arising from a weighted Lebesgue measure (a prototype is the multiplicative Hilbert matrix) and another from a discrete measure (a prototype is finite or infinite sum of rank one operators), and analyse the structural differences between them. We further relate the Schatten $p$-class membership of Helson matrices $mathbf a$ to the $ell^p$ membership of the sequence of norms of the associated analytic symbols $A_{n, mathbf a}$ and discuss applications to the spectral theory of Helson matrices.
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We consider reproducing kernel Hilbert spaces of Dirichlet series with kernels of the form $k(s,u) = sum a_n n^{-s-bar u}$, 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 $H^2_d$ in $d$ variables, where $d$ can be any number in ${1,2,ldots, infty}$, and in particular their multiplier algebras are unitarily equivalent to the multiplier algebra of $H^2_d$. Thus, a family of multiplier algebras of Dirichlet series are 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 to $H^2_d$ and when its multiplier algebra is isometrically isomorphic to $Mult(H^2_d)$.
Corresponding to any $(m-1)$-tuple of semi-spectral measures on the unit circle, a weighted Dirichlet-type space is introduced and studied. We prove that the operator of multiplication by the coordinate function on these weighted Dirichlet-type spaces acts as an analytic $m$-isometry and satisfies a certain set of operator inequalities. Moreover, it is shown that an analytic $m$-isometry which satisfies this set of operator inequalities can be represented as an operator of multiplication by the coordinate function on a weighted Dirichlet-type space induced from an $(m-1)$-tuple of semi-spectral measures on the unit circle. This extends a result of Richter as well as of Olofsson on the class of analytic $2$-isometries. We also prove that all left invertible $m$-concave operators satisfying the aforementioned operator inequalities admit a Wold-type decomposition. This result serves as a key ingredient to our model theorem and also generalizes a result of Shimorin on a class of $3$-concave 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 Tornehaves 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.
We consider composition operators $mathscr{C}_varphi$ on the Hardy space of Dirichlet series $mathscr{H}^2$, generated by Dirichlet series symbols $varphi$. We prove two different subordination principles for such operators. One concerns affine symbols only, and is based on an arithmetical condition on the coefficients of $varphi$. The other concerns general symbols, and is based on a geometrical condition on the boundary values of $varphi$. Both principles are strict, in the sense that they characterize the composition operators of maximal norm generated by symbols having given mapping properties. In particular, we generalize a result of J. H. Shapiro on the norm of composition operators on the classical Hardy space of the unit disc. Based on our techniques, we also improve the recently established upper and lower norm bounds in the special case that $varphi(s) = c + r2^{-s}$. A number of other examples are given.
Let $mathscr{H}^2$ denote the Hilbert space of Dirichlet series with square-summable coefficients. We study composition operators $mathscr{C}_varphi$ on $mathscr{H}^2$ which are generated by symbols of the form $varphi(s) = c_0s + sum_{ngeq1} c_n n^{-s}$, in the case that $c_0 geq 1$. If only a subset $mathbb{P}$ of prime numbers features in the Dirichlet series of $varphi$, then the operator $mathscr{C}_varphi$ admits an associated orthogonal decomposition. Under sparseness assumptions on $mathbb{P}$ we use this to asymptotically estimate the approximation numbers of $mathscr{C}_varphi$. Furthermore, in the case that $varphi$ is supported on a single prime number, we affirmatively settle the problem of describing the compactness of $mathscr{C}_varphi$ in terms of the ordinary Nevanlinna counting function. We give detailed applications of our results to affine symbols and to angle maps.
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