Hadamard multipliers on spaces of real analytic functions (Q390748)

The authors' aim is to find criteria for global solvability of an Euler differential equation NEWLINE\[NEWLINE\sum_{n=0}^{+\infty}a_n\theta^nf(t)=g(t),\quad t\in\mathbb{R},\,\,(a_n)_n\subset\mathbb{C}.NEWLINE\]NEWLINE Here, \(f,g\) are analytic functions on some open interval \(I\subset\mathbb{R}\). The main topic behind this theme is the notion of a multiplier. The fundamental question is surjectivity of a multiplier. The paper under review concentrates on the case \(0\in I\). Surjectivity for intervals not containing 0 is fully characterized in [\textit{M. Langenbruch} and \textit{P. Domański}, Analysis, München 32, No. 2, 137--162 (2012; Zbl 1288.46020)]. The challenge for \(0\in I\) comes from the fact that then Euler differential operators are singular. The paper fully describes multiplier sequences of holomorphic functions with restricted growth (Theorem 4.5). This allows the authors to completely characterize invertible multipliers (Corollaries 5.10, 5.11). In the case of Euler differential operators, it turns out that many perturbations of Euler differential operators are also invertible.