Summation of hyperharmonic series in Banach algebras and Banach modules

Abstract

By employing the Laplace transform for Banach-space-valued functions, in this paper we evaluate the sums of some hyperharmonic-like series in Banach algebras and modules. We discuss the cases when the general terms of the given series are invertible in the respective algebras, and when they
are invertible in the Drazin-Koliha sense, or the Mary-Patricio sense. Afterwards, we extend our results to the multilateral modular series of the form 
$$\Sum_{k=1}^{\infty}(a_1 + k)^{−n_1}c_1(a_2 + k)^{−n_2}c_2\cdot\ldots\cdot(a_{m−1} + k)^{−n_{m−1}}c_{m−1}(a_m + k)^{−n_m},$$
where $a_i$ belong to possibly different Banach algebras, $c_j$ belong to possibly different Banach bimodules, and $n_1,\ldots, n_m$ are positive integers. As an application, we obtain a new necessary solvability condition
for the Sylvester equation $ax−xb = c$ in Banach bimodules.