Banach spaces of general Dirichlet series
Fix a strictly increasing sequence \(\lambda = (\lambda_n)\) of positive real numbers which tends to infinity. The authors study general Dirichlet series of the form \(D=\sum a_n \lambda_n^s\), where the coefficients \((a_n)\) form a sequence of complex numbers, and \(s\) is a complex variable. Denote by \(\mathcal{H}_\infty(\lambda)\) the space of all such series which converge on the positive right half space \([\text{Re} >0]\) and which define there a bounded (and then necessarily holomorphic) function. The main interest of this article is to show that under the condition [\(\exists q >0: \inf (\lambda_{n+1}^q - \lambda_n^q )] >0\)], the space \(\mathcal{H}_\infty(\lambda)\) together with the norm \(\|D\|_\infty = \sup_{\text{Re}s>0} |\sum a_n \lambda_n^s| \) forms a Banach space. This generalizes the well-known case of ordinary Dirichlet series \(\sum a_n n^{-s}\). The authors observe that their condition on \(\lambda\) in fact is equivalent to a well-known condition of Bohr [\(\exists \ell \geq 0 \;\forall \delta > 0: (\log \lambda_{n+1} /\lambda_n )^{-1} = O (\lambda_n^{\ell + \delta})\)]. \textit{H. Bohr} [J. Reine Angew. Math. 143, 203--211 (1913; JFM 44.0307.01)] showed that under his condition on \(\lambda\) for each \(D=\sum a_n \lambda_n^s\) which converges at least in some \(s\), the following two facts hold true: (1) \(D\) converges absolutely on some appropriate half space. (2) If the function defined by \(D\) extends to a bounded and holomorphic function on \([\text{Re} > 0]\), then \(D\) converges uniformly on each smaller half space \([\text{Re} > \varepsilon]\). In the spirit of Bohr's approach the authors provide another proof of these two facts which then in an almost straightforward way lead to the desired completeness of \(\mathcal{H}_\infty(\lambda)\). Additionally, it is shown that under certain (quite restrictive) further conditions on \(\lambda\) the Banach space \(\mathcal{H}_\infty(\lambda)\) can be isometrically identified with the Banach space \(H_\infty(B_{c_0})\) of all bounded holomorphic functions on the open unit ball of radius \(c_0\). Again this extends a result on ordinary Dirichlet series due to \textit{H. Hedenmalm} et al. [Duke Math. J. 86, No. 1, 1--37 (1997; Zbl 0887.46008)]. Finally, estimates for the maximal width of the strips on which general Dirichlet series converge (converge uniformly), but do not converge absolutely, are given.
scientific article; zbMATH DE number 6898333
scientific article