Isometric shifts on C0(X)

Disjointness preserving shifts on C0(X)

“…With the new tools provided in [10] about Fredholm disjointness preserving operators we can perform an extensive study in disjointness preserving shifts on C 0 (X). We obtain quite a complete theory analogous to those presented in [6,7,11]. In particular, we know that every disjointness preserving quasi-n-shift T on C 0 (X) can be written as a weighted composition operator Tf = h · f • ϕ, where h is a bounded continuous scalar function on X away from zero and ϕ is a homeomorphism from X onto X modulo finite subsets.…”

“…We shall see in Examples 7.4 and 7.5 that such a hope is indeed fruitless. When X does not contain isolated points, it is shown in [11] that every isometric quasi-n-shift on C 0 (X) is disjointness preserving. Therefore, Example 7.5 gives also an example of an isometric quasi-n-shift which cannot be written as a product of n isometric quasi-1-shifts.…”

“…See, for example, [3,4,[6][7][8][9]14]. Isometric (quasi-)n-shifts on general locally compact spaces are discussed in [11].…”

On the separability problem for isometric shifts on C(X)

“…Among them, we will mention for instance [1,3,5,6,16,17,[19][20][21][22] (see also references therein).…”

Finite codimensional isometries on spaces of vector-valued continuous functions

“…This result of Holsztyński was used in [11] (see also [2,4,9,10,12,14,16]) to classify linear isometries on C(X) whose range has codimension 1 as follows: Let T : C(X) −→ C(X) be a codimension 1 linear isometry. Then there exists a closed subset X 0 of X such that either (1) X 0 = X \ {p} where p is an isolated point of X, or (2) X 0 = X, and such that there exists a continuous map h of X 0 onto X and a function a ∈ C(X 0 ), |a| ≡ 1, such that (T f )(x) = a(x) · f (h(x)) for all x ∈ X 0 and all f ∈ C(X).…”