Generalized shifts on Banach spaces of continuous functions
There is a definition of generalized shift due to \textit{R. J. Holub} [Can. Math. Bull. 31, No. 1, 85--94 (1988; Zbl 0658.47032)]. An operator \(T\) on a Banach space \(X\) is a generalized shift if it is an isometry with the range of codimension one and satisfying \(\bigcap_{n} T^n (X) =\{0\}\). There are several open problems about the existence of these operators on certain Banach spaces. It is also unknown if \(K\) has to be separable assuming that the space \(C(K) \) has a shift. Here it is proved that if \(K\) is Eberlein compact or scattered and \(C(K)\) (the spaces of real or complex valued functions on \(K\)) has a shift, then \(K\) has to be separable. Stability properties are also studied. For instance, it is proved that there are Banach spaces \(X\) and \(Y\) with shifts such that \(X \oplus _\infty Y\) does not have a shift. The paper also contains unsolved problems.
scientific article; zbMATH DE number 1995589