Fixed points and invariant domains of expansive holomorphic maps in complex Banach spaces (Q1842227)

The paper discusses the fixed point problems for a continuous compact map \(f:\overline{\Omega}\to E\) defined on an open bounded subset \(\Omega\) of a complex Banach space \(E\) being holomorphic on \(\Omega\). Using the proved infinite-dimensional version of the Rouche theorem (which, by the way, is a well-known excercise in the Leray-Schauder degree theory -- comp. the book of \textit{N. G. Lloyd} ``Degree theory'' [Cambridge University Press (1978; Zbl 0367.47001)] where it was proved for \(E=C^n\); but the proof provided in the paper is a verbatim repetition of the Lloyd's argument), the author studies the existence of fixed points and invariant domains for such a map \(f\) being expansive in the sense that \(f(\Omega)\supset \Omega\) and \(f(\overline{\Omega})\supset \overline{\Omega}\). The first part of the second theorem of the paper is a straightforward implication of the Rouche theorem; while the second one relies on a very strong assumption that \(I+f\) is compact which implies that \(\dim E<\infty\). The third result is an application of the second one in the context of an expansive holomorphic map \(f\) defined on an open ball \({\mathcal U}_0\) of a \(J^*\)-algebra \({\mathcal U}\). Again one of the assumptions is that \(I+f\) is compact, i.e. \({\mathcal U}\) is finite-dimensional. Thus, this restricts seriously the significance of the theorem and, at least, requires a comment which, unfortunately, is missing.