Ring homomorphisms on H(G)
Almost half a century ago, \textit{L. Bers} [Bull. Am. Math. Soc. 54, 311-315 (1948; Zbl 0032.20302)] showed that the rings H(G) and \(H(G')\) of analytic functions on planar domains G and \(G'\) are isomorphic if and only if G and \(G'\) are conformally equivalent. In particular, every normed ring automorphism \(\Phi\) of H(G) (that is one satisfying \(\Phi (i)=i)\) has the form \(\Phi (f)=f\circ h\) for some conformal mapping h of G onto G where \(h=\Phi (z)\). In the paper under review the author studies ring endomorphism of H(G) into H(G). He shows that if G satisfies \(G=\bar G^ 0\), then any ring endomorphism M of H(G) is linear (resp. conjugate linear), provided that M takes the identity function to a nonconstant function. In particular, if \(M(i)=i\), then M has the form \(M(f)=f\circ h\) for some function \(h\in H(G)\) which maps G into G \((h=M(z))\). The proof depends on an interpolation theorem of \textit{J. W. Nienhuys} and \textit{J. G. F. Thiemann} [Proc. Kon. Ned. Akad. Wetensch. 79, 331-334 (1976; Zbl 0336.30009)].
scientific article; zbMATH DE number 4163525