Backward shifts on function algebras
In [Canad. Math. Bull. 31, 85--94 (1988; Zbl 0658.47032)], \textit{J. R. Holub} introduced the concept of backward shifts on Banach spaces. In [J. Math. Anal. Appl. 202, 485--491 (1996; Zbl 0878.47014)], \textit{M. Rajagopalan} and \textit{K. Sundaresan} proved that \(C(X)\), the Banach space of all continuous function on a compact Hausdorff space \(X\) equipped with the uniform norm, does not admit a backward shift if \(X\) is infinite. As a generalization, the authors in this paper show that an infinite-dimensional function algebra \(A\) on \(X\) which is a uniformly closed subalgebra of \(C(X)\) containing the constants and separating points of \(X\), does not admit a backward shift. Since a finite-dimensional space always admits a backward shift, the adjective ``infinite-dimensional'' is necessary here. Note that backward shifts on finite-dimensional space are not surjective, however, they are always surjective on infinite-dimensional spaces (see Proposition~1.2 in [\textit{M. Rajagopalan} and \textit{K. Sundaresan}, op.\,cit.]). Moreover, they define a backward quasi-shift as a weak type of a backward shift, and show that a function algebra \(A\) does not admit a surjective backward quasi-shift under the assumption that the Choquet boundary of \(A\) has at most finitely many isolated points. There are examples given in the end of the paper.
scientific article; zbMATH DE number 5854864