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Quasi-Invariance of the Dirichlet series kernels, Analytic symbols and Homogeneous operators

For a scalar matrix $\mathbf a=(a_{m, n})_{m, n=1}^{\infty},$ the Dirichlet series kernel $κ_{\mathbf a}$ is the double Dirichlet series $κ_{\mathbf a}(s, u) = \sum_{m, n =1}^{\infty} a_{m, n}m^{-s} n^{-\bar{u}}$ in the variables $s$ and $\bar{u},$ which is regularly convergent on some right half-plane $\mathbb H_ρ.$ The analytic symbols $A_{n, \mathbf a} = \sum_{m=1}^{\infty}a_{m, n}m^{-s},$ $n \geq 1$ play a central role in the study of the reproducing kernel Hilbert space $\mathscr H_{\mathbf a}$ associated with the positive semi-definite kernel $κ_{\mathbf a}.$ In particular, they form a total subset of $\mathscr H_{\mathbf a}$ and provide the formula $\sum_{n=1}^{\infty}\langle f, A_{n, \mathbf a} \rangle n^{-s},$ $s \in \mathbb H_ρ,$ for $f \in \mathscr H_{\mathbf a}.$ We combine the basic theory of Dirichlet series kernels with the Gelfond-Schneider theorem (Hilbert's seventh problem) to show that any quasi-invariant Dirichlet series kernel $κ_{\mathbf a}(s, u)$ factors as $f(s)\bar{f(u)}$ for some Dirichlet series $f$ on $\mathbb H_ρ.$ In particular, there is no quasi-invariant Dirichlet series kernel $κ_{\mathbf a}$ if the dimension of $\mathscr H_{\mathbf a}$ is bigger than one. This is in strict contrast with the case of the unit disc, where non-factorable quasi-invariant kernels exist in abundance. We further discuss the Dirichlet series kernels $κ_{\mathbf a}$ invariant under the group $\mathscr T$ of translation automorphisms of $\mathbb H_ρ$ and construct a family of densely defined $\mathscr T$-homogeneous operators in $\mathscr H_{\mathbf a},$ whose adjoints are defined only at the zero vector.