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Subnormality of Weighted Shifts Associated to Composition Operators

Subnormality of Weighted Shifts Associated to Composition Operators Let $$\varphi $$ φ be a linear fractional transformation mapping the unit disk $$\mathbb {D}$$ D into itself and fixing 1, and let $$C_\varphi $$ C φ be the usual composition operator induced by $$\varphi $$ φ on the Hardy space $$H^2(\mathbb {D})$$ H 2 ( D ) . For a point z of $$\mathbb {D}$$ D , the subspace generated by the reproducing kernels $$k_{\varphi ^{n}(z)}$$ k φ n ( z ) ( $$n= 0, 1, \ldots $$ n = 0 , 1 , … ) arising from iterates of z under $$\varphi $$ φ is invariant for the adjoint $$C^*_\varphi $$ C φ ∗ . On this subspace, we show that $$C^*_\varphi $$ C φ ∗ is similar to a weighted shift, and the subnormality and hyponormality of the shift or their lack (which, surprisingly, coincide) are determined by z and the location of the other fixed point of $$\varphi $$ φ in pleasing geometrical ways. These results are applied to semigroups of composition operators arising from linear fractional transformations. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Subnormality of Weighted Shifts Associated to Composition Operators

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Publisher
Springer Journals
Copyright
Copyright © 2019 by Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-019-00268-x
Publisher site
See Article on Publisher Site

Abstract

Let $$\varphi $$ φ be a linear fractional transformation mapping the unit disk $$\mathbb {D}$$ D into itself and fixing 1, and let $$C_\varphi $$ C φ be the usual composition operator induced by $$\varphi $$ φ on the Hardy space $$H^2(\mathbb {D})$$ H 2 ( D ) . For a point z of $$\mathbb {D}$$ D , the subspace generated by the reproducing kernels $$k_{\varphi ^{n}(z)}$$ k φ n ( z ) ( $$n= 0, 1, \ldots $$ n = 0 , 1 , … ) arising from iterates of z under $$\varphi $$ φ is invariant for the adjoint $$C^*_\varphi $$ C φ ∗ . On this subspace, we show that $$C^*_\varphi $$ C φ ∗ is similar to a weighted shift, and the subnormality and hyponormality of the shift or their lack (which, surprisingly, coincide) are determined by z and the location of the other fixed point of $$\varphi $$ φ in pleasing geometrical ways. These results are applied to semigroups of composition operators arising from linear fractional transformations.

Journal

Computational Methods and Function TheorySpringer Journals

Published: May 3, 2019

References