On locally bounded maps of sequence spaces (Q797090)

A note of \textit{W. B. Jurkat}, \textit{B. L. R. Shawyer} [Analysis 1, 209-210 (1981; Zbl 0512.46004)] is slightly corrected and generalized. The Jurkat-Shawyer's theorem says that if A and B are two summation methods for (complex) sequences such that: \(\lim_ A1=\lim_ B1=1, \lim_ A(- 1)^ n=\lim_ B(-1)^ n=\theta \in C,\) and if \(f:\quad C\to C\) is continuous at at least one point, then f has the property P (i.e. \(\lim_ Bf(x_ n)=f(\lim_ ax_ n)\) for every A-summable sequence \((x_ n))\) if and only if there exist a,b,\(c\in C\) such that \(f(z)=az+b\bar z+c.\) The ''correction'' concernes the hypothesis \(\theta \in C\); \(\theta\) ''must be in R''. This theorem is now generalized by the following. ''Let A and B satisfy: \(\lim_ A(-1)^ n=\theta, \lim_ B(-1)^ n=\omega, \lim_ A1=\phi, \lim_ B1=\psi, \theta \neq \pm \phi, \omega \neq \pm \psi,\) let \(f:C\to C\) be bounded on a set of positive measure and let f have the property P. Then there exist a,b,\(c\in C\) such that \(f(z)=az+b\bar z+c, \forall z\in C\). Conversely, if \(f(z)=z\), \(f(z)=\bar z\) or \(f(z)=1\) have the property P, then \(\phi =\psi\), \(\theta =\omega\) or \(\phi ={\bar \psi}, \theta ={\bar \omega}\) or \(\psi =1\). If and only if \(\phi =\psi =1\), \(\theta =\omega \in R\) then every function of the form \(f(z)= az+b\bar z+c\) has the property P''. The corresponding real case is also considered.