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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 $Mult(H^2_d)$.
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In this article we examine Dirichlet type spaces in the unit polydisc, and multipliers between these spaces. These results extend the corresponding work of G. D. Taylor in the unit disc. In addition, we consider functions on the polydisc whose restrictions to lower dimensional polydiscs lie in the corresponding Dirichet type spaces. We see that such functions need not be in the Dirichlet type space of the whole polydisc. Similar observations are made regarding multipliers.
This paper is devoted to the study of reducing subspaces for multiplication operator $M_phi$ on the Dirichlet space with symbol of finite Blaschke product. The reducing subspaces of $M_phi$ on the Dirichlet space and Bergman space are related. Our strategy is to use local inverses and Riemann surface to study the reducing subspaces of $M_phi$ on the Bergman space, and we discover a new way to study the Riemann surface for $phi^{-1}circphi$. By this means, we determine the reducing subspaces of $M_phi$ on the Dirichlet space when the order of $phi$ is $5$; $6$; $7$ and answer some questions of Douglas-Putinar-Wang cite{DPW12}.
Given scalars $a_n ( eq 0)$ and $b_n$, $n geq 0$, the tridiagonal kernel or band kernel with bandwidth $1$ is the positive definite kernel $k$ on the open unit disc $mathbb{D}$ defined by [ k(z, w) = sum_{n=0}^infty Big((a_n + b_n z)z^nBig) Big((bar{a}_n + bar{b}_n bar{w}) bar{w}^n Big) qquad (z, w in mathbb{D}). ] This defines a reproducing kernel Hilbert space $mathcal{H}_k$ (known as tridiagonal space) of analytic functions on $mathbb{D}$ with ${(a_n + b_nz) z^n}_{n=0}^infty$ as an orthonormal basis. We consider shift operators $M_z$ on $mathcal{H}_k$ and prove that $M_z$ is left-invertible if and only if ${|{a_n}/{a_{n+1}}|}_{ngeq 0}$ is bounded away from zero. We find that, unlike the case of weighted shifts, Shimorins models for left-invertible operators fail to bring to the foreground the tridiagonal structure of shifts. In fact, the tridiagonal structure of a kernel $k$, as above, is preserved under Shimorin model if and only if $b_0=0$ or that $M_z$ is a weighted shift. We prove concrete classification results concerning invariance of tridiagonality of kernels, Shimorin models, and positive operators. We also develop a computational approach to Aluthge transforms of shifts. Curiously, in contrast to direct kernel space techniques, often Shimorin models fails to yield tridiagonal Aluthge transforms of shifts defined on tridiagonal spaces.
We prove an analogue of the Lagrange Inversion Theorem for Dirichlet series. The proof is based on studying properties of Dirichlet convolution polynomials, which are analogues of convolution polynomials introduced by Knuth in [4].
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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