Analytic functions of topological proper contractions (Q1072079)

The authors prove that if \(\{A_ 1,...,A_ N\}\) is a finite system of pairwise commuting operators on a Hilbert space \({\mathcal H}\) with spectral radii \(r(A_ i)<1\), then these operators are simultaneous topological proper contractions, i.e., there is a new Hilbert norm \(\| \cdot \|_*\) on \({\mathcal H}\), equivalent to the original one, such that \(\| A_ i\|_*<1\) for all \(i=1,...,N.\) As an application the authors establish analogues of some results of \textit{Ky Fan} [ibid. 160, 275-290 (1978; Zbl 0441.30060)] for analytic functions of topological proper contractions.