Essential Norm of Composition Operators on Banach Spaces of Hölder Functions
Let (X,d) be a pointed compact metric space, let 0<α<1, and let φ:X→X be a base point preserving Lipschitz map. We prove that the essential norm of the composition operator Cφ induced by the symbol φ on the spaces lip0(X,dα) and Lip0(X,dα) is given by the formula ‖Cφ‖e=limt→0 sup0<d(x, y)&...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2011-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2011/590853 |
| Summary: | Let (X,d) be a pointed compact metric space, let 0<α<1, and let φ:X→X be a base point preserving Lipschitz map. We prove that the essential norm of the composition operator Cφ induced by the symbol φ on the spaces lip0(X,dα) and Lip0(X,dα) is given by the formula ‖Cφ‖e=limt→0 sup0<d(x, y)<t(d(φ(x),φ(y))α/d(x,y)α) whenever the dual space lip0(X,dα)∗ has the approximation property. This happens in particular when X is an infinite compact subset of a finite-dimensional normed linear space. |
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| ISSN: | 1085-3375 1687-0409 |