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On circular Kippenhahn curves and the Gau-Wang-Wu conjecture about nilpotent partial isometries

Eric Shen

Abstract

We study linear operators on a finite-dimensional space whose Kippenhahn curves consist of concentric circles centered at the origin. We say that such operators have Circularity property. One class of examples is rotationally invariant operators. To every operator with norm at most one, we associate an infinite sequence of partial isometries and study when Circularity property can be passed back and forth along that sequence. In particular, we introduce a class of operators for which every partial isometry in the aforementioned sequence has Circularity property, and show that this class is broader than the class of rotationally invariant operators. As a consequence, every such an operator provides a counterexample to the Gau--Wang--Wu conjecture about nilpotent partial isometries. We also discuss possible refinements of the conjecture. Finally, we propose a way to check whether a matrix is rotationally invariant, suitable for numerical experiments.

On circular Kippenhahn curves and the Gau-Wang-Wu conjecture about nilpotent partial isometries

Abstract

We study linear operators on a finite-dimensional space whose Kippenhahn curves consist of concentric circles centered at the origin. We say that such operators have Circularity property. One class of examples is rotationally invariant operators. To every operator with norm at most one, we associate an infinite sequence of partial isometries and study when Circularity property can be passed back and forth along that sequence. In particular, we introduce a class of operators for which every partial isometry in the aforementioned sequence has Circularity property, and show that this class is broader than the class of rotationally invariant operators. As a consequence, every such an operator provides a counterexample to the Gau--Wang--Wu conjecture about nilpotent partial isometries. We also discuss possible refinements of the conjecture. Finally, we propose a way to check whether a matrix is rotationally invariant, suitable for numerical experiments.

Paper Structure

This paper contains 5 sections, 24 theorems, 197 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Partial isometry towers and defect-pencil Circularity
  4. A relaxation of the Gau-Wang-Wu conjecture
  5. Computable criteria for rotational invariance

Key Result

Proposition 2.2

Let $A$ be a partial isometry with nontrivial kernel. Then, after a unitary similarity, where $m=\dim\ker A$, $d=\dim(\ker A)^\perp$, and Conversely, every matrix of this form satisfying eq:defect-relation is a partial isometry.

Figures (2)

  • Figure 1: The Kippenhahn curve of a partial isometry from Example \ref{['ex:P-not-Q']}. Red dots are placed at $(-0.4, 0)$ and $(0.4, 0)$, to highlight that the green branch is not a circle.
  • Figure 2: The Kippenhahn curve of a nilpotent partial isometry with circular numerical range without Circularity property. The outer branch is a circle, hence the numerical range is a disc.

Theorems & Definitions (54)

  • Definition 1.1
  • Conjecture 1.2: Gau--Wang--Wu
  • Definition 2.1
  • Proposition 2.2: Canonical form of a partial isometry
  • Theorem 2.3: GWW, Theorem 4.7
  • Proposition 2.4: GWW, Proposition 2.6
  • Definition 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • ...and 44 more