eBook ISBN:  9781470403898 
Product Code:  MEMO/167/791.E 
List Price:  $60.00 
MAA Member Price:  $54.00 
AMS Member Price:  $36.00 
eBook ISBN:  9781470403898 
Product Code:  MEMO/167/791.E 
List Price:  $60.00 
MAA Member Price:  $54.00 
AMS Member Price:  $36.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 167; 2004; 81 ppMSC: Primary 47; Secondary 30
A bounded operator \(T\) acting on a Hilbert space \(\mathcal H\) is called cyclic if there is a vector \(x\) such that the linear span of the orbit \(\{T^n x : n \geq 0 \}\) is dense in \(\mathcal H\). If the scalar multiples of the orbit are dense, then \(T\) is called supercyclic. Finally, if the orbit itself is dense, then \(T\) is called hypercyclic. We completely characterize the cyclicity, the supercyclicity and the hypercyclicity of scalar multiples of composition operators, whose symbols are linear fractional maps, acting on weighted Dirichlet spaces. Particular instances of these spaces are the Bergman space, the Hardy space, and the Dirichlet space. Thus, we complete earlier work on cyclicity of linear fractional composition operators on these spaces. In this way, we find exactly the spaces in which these composition operators fail to be cyclic, supercyclic or hypercyclic. Consequently, we answer some open questions posed by Zorboska. In almost all the cases, the cutoff of cyclicity, supercyclicity or hypercyclicity of scalar multiples is determined by the spectrum. We will find that the Dirichlet space plays a critical role in the cutoff.
ReadershipGraduate students and research mathematicians interested in operator theory.

Table of Contents

Chapters

1. Introduction and preliminaries

2. Linear fractional transformations with an interior or an exterior fixed point

3. Non elliptic automorphisms

4. The parabolic non automorphism

5. Supercyclic linear fractional composition operators


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A bounded operator \(T\) acting on a Hilbert space \(\mathcal H\) is called cyclic if there is a vector \(x\) such that the linear span of the orbit \(\{T^n x : n \geq 0 \}\) is dense in \(\mathcal H\). If the scalar multiples of the orbit are dense, then \(T\) is called supercyclic. Finally, if the orbit itself is dense, then \(T\) is called hypercyclic. We completely characterize the cyclicity, the supercyclicity and the hypercyclicity of scalar multiples of composition operators, whose symbols are linear fractional maps, acting on weighted Dirichlet spaces. Particular instances of these spaces are the Bergman space, the Hardy space, and the Dirichlet space. Thus, we complete earlier work on cyclicity of linear fractional composition operators on these spaces. In this way, we find exactly the spaces in which these composition operators fail to be cyclic, supercyclic or hypercyclic. Consequently, we answer some open questions posed by Zorboska. In almost all the cases, the cutoff of cyclicity, supercyclicity or hypercyclicity of scalar multiples is determined by the spectrum. We will find that the Dirichlet space plays a critical role in the cutoff.
Graduate students and research mathematicians interested in operator theory.

Chapters

1. Introduction and preliminaries

2. Linear fractional transformations with an interior or an exterior fixed point

3. Non elliptic automorphisms

4. The parabolic non automorphism

5. Supercyclic linear fractional composition operators