Summation techniques in Banach algebras and modules, with applications

Abstract

This contribution explores advanced summability methods in the setting of Banach algebras and Banach modules, with a particular emphasis on their application to the study of ’hyperharmonic’ series. Lecture firstly gives a comparative overview of Laplace transforms in real space setting, Banach spaces, Banach algebras and Banach modules, as well as analysis of procedure proposed earlier by Gautchi and Milovanovic. Through the lens of Laplace transform and within the mentioned framework, we establish a connection with the hypergeometric and polygamma functions, thus lifting some known scalar series identities to this abstract setting. Moreover, we extend our results to the multilateral modular series, having the form $$\Sum \limits_{k=1} ^\infty (a_1+k)^{-{n_1}} c_1 \cdot (a_2+k)^{-{n_2}} c_2 \cdot \ldots \cdot (a_{m-1}+k)^{-{n_{m-1}}} c_{m-1} \cdot (a_m+k)^{-{n_m}}$$ where $a_i$ belong to possibly different Banach algebras, and $c_j$ belong to possibly different Banach bimodules, and $n_1,\ldots, n_m$ are positive integers. We obtain sums for series of the form $$\sum \limits_{k=1}^{\infty} (a+k)^{-n} , \sum \limits_{k=1}^{\infty} (-1)^k (a+k)^{-n} , \sum \limits_{k=1}^{\infty} ((a+k)^\Dagger)^{n}$$ where $\Dagger$ represents the Drazin-Koliha invertibility, or the invertibility along an idempotent. As an application, we obtain a new necessary solvability condition for the Sylvester equation ax−xb = c in Banach modules. Finally, the connection to C0−semigroups is given.