SHIFT-OPERATORS ON BANACH-SPACES

J. R. Holub has obtained several results for shift operators on C(X). In this paper we answer some questions of Holub and obtain extensions of many of his results. In particular, we show that C(X, R) does not admit a shift operator if X has only countably many components and each component is infinite. We show that C(X, C) does not admit a shift operator for certain special compact Hausdorff spaces X. We show that there exists a compact Hausdorff space X which is not totally disconnected and both C(X, C) and C(X, R) admit shift operators. If 1 ≤ p < ∞ and (X, Σ, μ) is a σ-finite non-atomic measure space then LPR(μ) does not admit a disjointness preserving shift operator. We also show that lP for 1 ≤ p ≤ ∞ is the only LPR(μ) space which admits a disjointness preserving shift operator. © 1991.