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The elliptical range theorem for the conformal range

Gyula Lakos

Abstract

The conformal range (or the real Davis--Wielandt shell), which is a particular planar projection of the Davis--Wielandt shell, can be considered as the hyperbolic version of the numerical range; i. e. it is a ``field of values'' which can be interpreted as a subset of the asymptotically closed hyperbolic plane. Here we explain the analogue of the elliptical range theorem of $2\times2$ complex matrices for the conformal range.

The elliptical range theorem for the conformal range

Abstract

The conformal range (or the real Davis--Wielandt shell), which is a particular planar projection of the Davis--Wielandt shell, can be considered as the hyperbolic version of the numerical range; i. e. it is a ``field of values'' which can be interpreted as a subset of the asymptotically closed hyperbolic plane. Here we explain the analogue of the elliptical range theorem of complex matrices for the conformal range.
Paper Structure (21 sections, 31 theorems, 167 equations)

This paper contains 21 sections, 31 theorems, 167 equations.

Table of Contents

  1. On the Davis--Wielandt shell
  2. On the hyperbolic space (review)
  3. The Davis--Wielandt shell (review)
  4. The elliptical range theorem for the shell (statement)
  5. The conformal range
  6. On the hyperbolic plane (review)
  7. The conformal range
  8. The qualitative elliptical range theorem for the conformal range
  9. On $2\times2$ complex matrices
  10. The spectral and metric discriminants
  11. The canonical representatives (complex Möbius)
  12. Spectral type refined
  13. Another spectral quantity
  14. A remark on the real double
  15. The canonical representatives (real Möbius)
  16. ...and 6 more sections

Key Result

Theorem 1.1

If $U$ is unitary, then $\mathop{\mathrm{DW}}\nolimits_{*}(A)=\mathop{\mathrm{DW}}\nolimits_{*}(UAU^{-1})$.

Theorems & Definitions (70)

  • Theorem 1.1: Wielandt, Davis
  • Theorem 1.2: Wielandt, Davis
  • Theorem 1.3: Wielandt, Davis
  • proof : Proofs
  • Theorem 1.4: Wielandt, Davis
  • Theorem 1.5: Addendum to Theorem \ref{['thm:DWdonc']}, on the inner metrical data
  • Theorem 1.6: Addendum to Theorem \ref{['thm:DWdonc']}, using "external" data
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • ...and 60 more