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The Role of the Spectrum in the Cyclic Behavior of Composition Operators
 
Eva A. Gallardo-Gutiérrez University of Zaragoza, Zaragoza, Spain
Alfonso Montes-Rodríguez University of Seville, Seville, Spain
The Role of the Spectrum in the Cyclic Behavior of Composition Operators
eBook ISBN:  978-1-4704-0389-8
Product Code:  MEMO/167/791.E
List Price: $60.00
MAA Member Price: $54.00
AMS Member Price: $36.00
The Role of the Spectrum in the Cyclic Behavior of Composition Operators
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The Role of the Spectrum in the Cyclic Behavior of Composition Operators
Eva A. Gallardo-Gutiérrez University of Zaragoza, Zaragoza, Spain
Alfonso Montes-Rodríguez University of Seville, Seville, Spain
eBook ISBN:  978-1-4704-0389-8
Product Code:  MEMO/167/791.E
List Price: $60.00
MAA Member Price: $54.00
AMS Member Price: $36.00
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1672004; 81 pp
    MSC: 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 cut-off 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 cut-off.

    Readership

    Graduate 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1672004; 81 pp
MSC: 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 cut-off 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 cut-off.

Readership

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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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