Disjointness preserving shifts on \(C_{0}(x)\)
type: Item
instance of: scholarly article
title: Disjointness preserving shifts on \(C_{0}(x)\)
zbMATH Open document ID: 1108.47029
author: N. E. Zubov
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-12-07T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 47B37
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 47A53
zbMATH DE Number: 5078935
zbMATH Keywords: shift operator
zbMATH Keywords: quasi-shift
zbMATH Keywords: disjointness preserving operator
zbMATH Keywords: Fredholm composition operator
reviewed by: Oleksiy Karlovych
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.01.067
OpenAlex ID: W2025100113
cites work: Q3308615
cites work: Q3683058
cites work: Codimension 1 linear isometries on function algebras
cites work: Isometric shifts and metric spaces
cites work: Commutants of shifts on Banach spaces
cites work: Isometric Shift Operators on C(X)
cites work: Shift operators on Banach spaces
cites work: Isometric shifts on \({\mathcal C}(K)\)
cites work: On Shift Operators
cites work: Q4422049
cites work: Isometric shifts on \(C _{0}(X)\).
cites work: Concrete representation of abstract (M)-spaces. (A characterization of the space of continuous functions.)
cites work: Q3713006
cites work: Backward shifts on Banach spaces \(C(X)\)
cites work: Q5678200
DOI: 10.1016/J.JMAA.2006.01.067
description: scientific article; zbMATH DE number 5078935