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On Kippenhahn curves of low rank partial isometries

Nikita Popov, Eric Shen, Ilya M. Spitkovsky

TL;DR

This work analyzes when rank-low partial isometries have Kippenhahn curves $C(A)$ containing circular components, thereby causing circular numerical ranges. Using unitary invariance, canonical block forms, and explicit Kippenhahn polynomial criteria, the authors derive precise conditions for the existence and location of circular components, with all such circles necessarily centered at the origin for rank-three cases. They provide a complete defect-based classification (defect $\operatorname{def}A=0,1,2$) and give explicit normal forms, parametric families, and geometric consequences (disks, ellipses, and multi-circle configurations). A central outcome is that the Gau-Wang-Wu conjecture holds for rank-three partial isometries, linking circle components to origin-centered disks, and they illustrate a spectrum of Kippenhahn curve geometries via examples. The results deepen understanding of numerical ranges for finite-rank operators and connect algebraic curve methods with operator theory.

Abstract

Conditions are established for rank three partial isometries to have circular components contained in their Kippenhahn curves. In particular, such matrices with circular numerical ranges are described. It is also established that the Gau-Wang-Wu conjecture holds for matrices under consideration.

On Kippenhahn curves of low rank partial isometries

TL;DR

This work analyzes when rank-low partial isometries have Kippenhahn curves containing circular components, thereby causing circular numerical ranges. Using unitary invariance, canonical block forms, and explicit Kippenhahn polynomial criteria, the authors derive precise conditions for the existence and location of circular components, with all such circles necessarily centered at the origin for rank-three cases. They provide a complete defect-based classification (defect ) and give explicit normal forms, parametric families, and geometric consequences (disks, ellipses, and multi-circle configurations). A central outcome is that the Gau-Wang-Wu conjecture holds for rank-three partial isometries, linking circle components to origin-centered disks, and they illustrate a spectrum of Kippenhahn curve geometries via examples. The results deepen understanding of numerical ranges for finite-rank operators and connect algebraic curve methods with operator theory.

Abstract

Conditions are established for rank three partial isometries to have circular components contained in their Kippenhahn curves. In particular, such matrices with circular numerical ranges are described. It is also established that the Gau-Wang-Wu conjecture holds for matrices under consideration.
Paper Structure (10 sections, 35 theorems, 87 equations, 6 figures)

This paper contains 10 sections, 35 theorems, 87 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Finite rank partial isometries
  4. Gau-Wang-Wu's conjecture for rank three operators
  5. Kippenhahn curves containing circular components
  6. Nilpotent rank three partial isometries
  7. Matrices with Kippenhahn curves containing $\mathcal{C}_{1/2}$
  8. Rank three partial isometries of defect two
  9. Examples of $C(A)$ for $A$ with $\operatorname{rank} A=3, \operatorname{def} A=2$
  10. Acknowledgements

Key Result

Proposition 1.1

For a rank $k$ operator $A$ is a reducing subspace of dimension $n\leq 2k$, and the restriction $A|\mathcal{M}$ of $A$ onto $\mathcal{M}^\perp=\ker A\cap\ker A^*$ is the zero operator.

Figures (6)

  • Figure 1: $\sqrt{1-d^2}=0.2$ and $\sqrt{1-h^2}=0.3$
  • Figure 2: $\sqrt{1-d^2}=0.2$ and $\sqrt{1-h^2}=0.6$
  • Figure 3: $\sqrt{1-d^2}=0.2$ and $\sqrt{1-h^2}=0.99$
  • Figure 4: Only circle of radius $r \approx 0.48$
  • Figure 5: Only circle of radius $r \approx 0.41$.
  • ...and 1 more figures

Theorems & Definitions (62)

  • Proposition 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Proposition 2.5
  • Theorem 3.1
  • proof
  • ...and 52 more