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Examples of isometric shifts on \(C(X)\). (English) Zbl 1248.47034

Let \(X\) be a compact Hausdorff space and \(C(X)\) the Banach space of all \(\mathbb{K}\)-valued (\(\mathbb{K}= \mathbb{R}\) or \(\mathbb{C}\)) continuous functions endowed with the sup norm \(\|.\|_{\infty}\). A linear map \(T: C(X) \to C(X)\) is called an isometric shift if \(T\) is an isometry, the codimension of the image \(\mathrm{im}(T)\) of \(T\) in \(C(X)\) is 1 and \(\bigcap_{n=1}^{\infty} \mathrm{im}(T^n)= \{0\}\).
The general form of an isometric shift was given in [A. Gutek et al., J. Funct. Anal. 101, No. 1, 97–119 (1991; Zbl 0818.47028)]: there exist a closed subset \(Y \subset X\), a continuous and surjective map \(\phi: Y \to X\) and a function \(a \in C(Y)\), \(|a| \equiv 1\) with \((Tf)(x)=a(x) f(\phi(x))\) for every \(x \in Y\) and every \(f \in C(X)\). Therefore isometric shifts may be classified into two types. \(T\) is said to be of type I if \(Y\) can be chosen to be equal to \(X \backslash \{p\}\), where \(p\) is an isolated point, while we call \(T\) to be of type II if \(Y\) can be taken equal to \(X\).
In [J. Funct. Anal. 135, No. 1, 157–162 (1996; Zbl 0866.47019)], R. Haydon studied isometric shifts of type II thoroughly and provided a general method for obtaining isometric shifts. In the article under review, the author considers isometric shifts of type I. In fact, he gives two different methods for obtaining such shifts. The first one admits the construction of examples with infinitely many nonhomeomorphic components in any infinite-dimensional normed space, while the second technique yields several applications, e.g., to sequences adjoined to any \(n\)-dimensional compact manifold (for \(n \geq 2\)) or the Sierpinski curve. Finally, combining both methods, the author treats examples with special features involving a convergent sequence adjoined to the Cantor set.

MSC:

47B38 Linear operators on function spaces (general)
46E15 Banach spaces of continuous, differentiable or analytic functions
54C35 Function spaces in general topology
54H20 Topological dynamics (MSC2010)
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References:

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