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On singular pencils with commuting coefficients

Vadym Koval, Patryk Pagacz

Abstract

We investigate the relation between the spectrum of matrix (or operator) polynomials and the Taylor spectrum of its coefficients. We prove that the polynomial of commuting matrices is singular, i.e. its spectrum is the whole complex plane, if and only if (0, 0, ... , 0) belongs to the Taylor spectrum of its coefficients. On the other hand we prove that this equivalence is not longer true if we consider the operators on infinite dimensional Hilbert space as coefficients of polynomial. As a consequence we could propose a new description of (Taylor) spectrum of k-tuple of matrices and we could disprove the conjecture previously proposed in the literature. Additionally, we pointed out the Kronecker forms of the pencils with commuting coefficients.

On singular pencils with commuting coefficients

Abstract

We investigate the relation between the spectrum of matrix (or operator) polynomials and the Taylor spectrum of its coefficients. We prove that the polynomial of commuting matrices is singular, i.e. its spectrum is the whole complex plane, if and only if (0, 0, ... , 0) belongs to the Taylor spectrum of its coefficients. On the other hand we prove that this equivalence is not longer true if we consider the operators on infinite dimensional Hilbert space as coefficients of polynomial. As a consequence we could propose a new description of (Taylor) spectrum of k-tuple of matrices and we could disprove the conjecture previously proposed in the literature. Additionally, we pointed out the Kronecker forms of the pencils with commuting coefficients.
Paper Structure (5 sections, 8 theorems, 71 equations)

This paper contains 5 sections, 8 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Basic notions
  3. Matrix pencils and matrix polynomials
  4. Operator pencils
  5. Kronecker forms of pencils with commuting coefficients

Key Result

Theorem 2.1

(see G) Let $A,B\in \mathbb{C}^{n\times m}$. Then there exist invertible matrices $S\in \mathbb{C}^{n\times n}$ and $T \in\mathbb{C}^{m\times m}$ such that, where $\mathcal{L}_{t}\in \mathbb{C}^{t\times(t+1)}$ are bidiagonal matrices of the form $\mathcal{J}_{\gamma_i}^{\lambda_i}\in \mathbb{C}^{\gamma_i\times\gamma_i}$ are Jordan blocks of the form and $\mathcal{N}_{\beta_i}\in\mathbb{C}^{\bet

Theorems & Definitions (19)

  • Theorem 2.1
  • Theorem 3.1
  • proof
  • Example 3.1
  • Example 3.2
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Example 3.3
  • ...and 9 more