Disjointness preserving shifts on \(C_{0}(x)\)
A bounded linear operator \(S\) on a Banach space \(E\) is called a quasi-\(n\)-shift if it is injective, has closed range, and has corank \(n\). A quasi-\(n\)-shift \(S\) is called an \(n\)-shift if \(\bigcap_{m=1}^\infty S^mE=\{0\}\). Let \(X\) be a locally compact Hausdorff space. It is shown that the space \(C_0(X)\) admits a quasi-\(n\)-shift \(T\) if and only if there exists a countable subset of \(X_\infty:=X\cup\{\infty\}\) equipped with a tree-like structure, called \(\varphi\)-tree. If \(T\) is an \(n\)-shift, then this \(\varphi\)-tree is dense in \(X\). By using the notion of a \(\varphi\)-tree, the authors prove that every (quasi)-\(n\)-shift on the space \(c_0\) can always be written as a product of \(n\) (quasi)-\(1\)-shifts. It is shown that for arbitrary spaces \(C_0(X)\), this fact does not hold. However, after an appropriate dilation, (quasi)-\(n\)-shifts on \(C_0(X)\) can be represented as products of (quasi)-\(1\)-shifts.
scientific article; zbMATH DE number 5078935
scientific article