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تسجيل مستخدم جديدAnalytic 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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