Isomorphism classes of weighted spaces of holomorphic functions on some subsets of complex plane (Q2418548)

Let \(G\) be an open subset of the complex plane which is either a strip around the real axis, or the upper half plane or the complex plane minus the nonnegative real semi-axis. Let \(v\) be a strictly positive continuous function on \(G\), which is called a weight. Under some restrictive assumptions of the weight \(v\) in each case, the author proves that the Banach space \(H_v(G)\) of all holomorphic functions \(f\) on \(G\) such that \(v|f|\) is bounded in \(G\), is isomorphic to either \(\ell_{\infty}\) or \(H^{\infty}\). This result follows from an application of results due to \textit{W. Lusky} and the author [Math. Scand. 111, No. 2, 244--260 (2012; Zbl 1267.30111)] and of \textit{A. Harutyunyan} and \textit{W. Lusky} [Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Comput. 39, 125--135 (2013; Zbl 1289.46045)]. \par The main results about isomorphic classification of weighted Banach spaces of holomorphic functions are due to \textit{W. Lusky}; see, e.g., his important paper [Stud. Math. 175, No. 1, 19--45 (2006; Zbl 1114.46020)].