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Conjugacy classes of automorphisms of the unit ball in a complex Hilbert space

Rachna Aggarwal, Krishnendu Gongopadhyay, Mukund Madhav Mishra

Abstract

In this article, we consider the ball model of an infinite dimensional complex hyperbolic space, i.e. the open unit ball of a complex Hilbert space centered at the origin equipped with the Caratheodory metric. We consider the group of holomorphic automorphisms of the ball and classify the conjugacy classes of automorphisms. We also compute the centralizers for elements in the group of automorphisms.

Conjugacy classes of automorphisms of the unit ball in a complex Hilbert space

Abstract

In this article, we consider the ball model of an infinite dimensional complex hyperbolic space, i.e. the open unit ball of a complex Hilbert space centered at the origin equipped with the Caratheodory metric. We consider the group of holomorphic automorphisms of the ball and classify the conjugacy classes of automorphisms. We also compute the centralizers for elements in the group of automorphisms.
Paper Structure (12 sections, 45 theorems, 53 equations)

This paper contains 12 sections, 45 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Holomorphic automorphisms of $B$
  4. Some special sub-classes of $G$
  5. Conjugacy classes for elliptic and hyperbolic isometries
  6. Proof of Theorem \ref{['t1']}
  7. Conjugacy classes for parabolic isometries
  8. Stabilizer of infinity
  9. Parabolic isometries
  10. Centralizers of isometries
  11. Centralizers of hyperbolic isometries
  12. Centralizers of Heisenberg translations

Key Result

Proposition 1.1

MR A general element of $G$ is of the form $e^{i \theta} \left[ { } \right] \,\text{where}\,\theta \in \mathbb{R}$, $\xi \in H$, $a=\sqrt{1+\|\xi\|^2}$, $U \in \mathcal{U}(H)$ and A is a positive operator on $H$ defined by $A=I$ on $\left<\xi\right>^{\perp}$ and $A(\xi)=a\xi$.

Theorems & Definitions (77)

  • Proposition 1.1
  • Theorem 1.2: Elliptic isometry
  • Theorem 1.3: Hyperbolic isometry
  • Theorem 1.4: Parabolic isometry
  • Theorem 1.5: Centralizer of an elliptic isometry
  • Theorem 1.6: Centralizer of a hyperbolic isometry
  • Corollary 1.6.1
  • Corollary 1.6.2
  • Theorem 1.7: Centralizer of Heisenberg translation
  • Lemma 2.1
  • ...and 67 more