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On the elliptical range theorems for the Davis-Wielandt shell, the numerical range, and the conformal range

Gyula Lakos

Abstract

We consider the elliptical range theorems for the Davis--Wielandt shell, the numerical range, and the conformal range in terms of and related to their quadratic representations. The emphasis is on exposing a variety of elementary approaches.

On the elliptical range theorems for the Davis-Wielandt shell, the numerical range, and the conformal range

Abstract

We consider the elliptical range theorems for the Davis--Wielandt shell, the numerical range, and the conformal range in terms of and related to their quadratic representations. The emphasis is on exposing a variety of elementary approaches.
Paper Structure (44 sections, 77 theorems, 492 equations)

This paper contains 44 sections, 77 theorems, 492 equations.

Table of Contents

  1. Normality of $2\times2$ matrices and related concepts
  2. Spectral type (review)
  3. The metric discriminant (review)
  4. Canonical triangular form (review)
  5. The five data
  6. Normality (review)
  7. Invariance for complex Möbius transformations (review)
  8. The canonical representatives (complex Möbius) (review)
  9. Norms (review)
  10. The Davis--Wielandt shell of $2\times2$ matrices
  11. Complex asymptotic parametrization and $h$-collineations (review)
  12. The qualitative elliptical range theorems for the shell (review)
  13. Other models; distance (review)
  14. The matrix of the shell
  15. The dual viewpoint
  16. ...and 29 more sections

Key Result

Lemma 1.1

The five data eq:polr characterizes the complex $2\times2$ matrix $A$ up to conjugation by unitary matrices.

Theorems & Definitions (188)

  • Lemma 1.1
  • proof
  • Lemma 1.2
  • Remark 1.3
  • Lemma 1.4
  • Lemma 1.5
  • Example 1.6
  • Corollary 1.7
  • Lemma 1.8
  • Theorem 2.1: Wielandt, Davis
  • ...and 178 more