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The extended eigenvalues of composition operator on Bergman space and Fock space

Li Jiale

Abstract

A complex scalar k is said to be an extended eigenvalue of a bounded linear operator A on a complex Hilbert space if there is a nonzero operator X such that AX=kXA. There are some solutions to the problem of computing the extended eigenvalues for composition operators induced on the Fock space and Bergman space by linear fractional transformationsof the complex plane in this paper.

The extended eigenvalues of composition operator on Bergman space and Fock space

Abstract

A complex scalar k is said to be an extended eigenvalue of a bounded linear operator A on a complex Hilbert space if there is a nonzero operator X such that AX=kXA. There are some solutions to the problem of computing the extended eigenvalues for composition operators induced on the Fock space and Bergman space by linear fractional transformationsof the complex plane in this paper.
Paper Structure (17 sections, 30 theorems, 60 equations)

This paper contains 17 sections, 30 theorems, 60 equations.

Table of Contents

  1. introduction
  2. total discussion
  3. Extended eigenvalues and extended eigenoperators
  4. Composition operators on Hardy, Fock, Bergman Spaces
  5. classification of linear fractional transformation
  6. quasi-differential operator and quasi-integral operator
  7. preliminary results
  8. Extended eigenvalues on Fock spaces
  9. Elliptic automophism
  10. Other types
  11. Extended eigenvalues on Bergman spaces $L^{2}_a$
  12. Elliptic automorphism
  13. loxodromic map or a hyperbolic nonautomorphism of the third kind
  14. hyperbolic automorphism
  15. hyperbolic nonautomorphism type one
  16. ...and 2 more sections

Key Result

Lemma 2.1

Operator $A\in{B(H)}$ , then it is injective if and only if $0\notin{Ext(A)} .$

Theorems & Definitions (30)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Lemma 2.9
  • Lemma 2.10
  • ...and 20 more