Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The fundamental equations for inversion of operator pencils on Banach space

Amie Albrecht, Phil Howlett, Charles Pearce

TL;DR

We address the inversion of linear operator pencils $A(z)=A_0+A_1 z$ on Banach spaces, characterizing analytic resolvents on annuli ${\mathcal{U}}_{s,r}$ via a Laurent expansion $R(z)=\sum_{j\in \mathbb Z} R_j z^j$ whose coefficients satisfy left and right fundamental equations with geometric bounds. A closed-form resolvent $R(z)=(I z + R_{-1}A_0)^{-1}R_{-1}+(I+R_0A_1 z)^{-1}R_0$ is derived, along with a spectral-projection framework that yields a clean separation of the spectrum into interior and exterior components, and extends to polynomial pencils. The work also develops a thorough treatment of isolated singularities, including finite-order poles via Jordan chains and the possibility of isolated essential singularities, with Gaussian-elimination-based solution procedures for the determining equations in the matrix/pencil setting. Unbounded operators are addressed through Sobolev-type renorming to bounded operators, and the global structure of the resolvent is described via a direct-sum decomposition of the underlying spaces and a block-resolvent decomposition. Collectively, the results generalize existing matrix-pencil theory to Banach-space pencils and provide concrete tools for spectral analysis, perturbation theory, and applications to control and stochastic problems.

Abstract

We prove that the resolvent of a linear operator pencil is analytic on an open annulus if and only if the coefficients of the Laurent series satisfy a system of fundamental equations and are geometrically bounded. Our analysis extends earlier work on the fundamental equations to include the case where the resolvent has an isolated essential singularity. We find a closed form for the resolvent and use the fundamental equations to establish key spectral separation properties when the resolvent has only a finite number of isolated singularities. Finally we show that our results can also be applied to polynomial pencils.

The fundamental equations for inversion of operator pencils on Banach space

TL;DR

We address the inversion of linear operator pencils on Banach spaces, characterizing analytic resolvents on annuli via a Laurent expansion whose coefficients satisfy left and right fundamental equations with geometric bounds. A closed-form resolvent is derived, along with a spectral-projection framework that yields a clean separation of the spectrum into interior and exterior components, and extends to polynomial pencils. The work also develops a thorough treatment of isolated singularities, including finite-order poles via Jordan chains and the possibility of isolated essential singularities, with Gaussian-elimination-based solution procedures for the determining equations in the matrix/pencil setting. Unbounded operators are addressed through Sobolev-type renorming to bounded operators, and the global structure of the resolvent is described via a direct-sum decomposition of the underlying spaces and a block-resolvent decomposition. Collectively, the results generalize existing matrix-pencil theory to Banach-space pencils and provide concrete tools for spectral analysis, perturbation theory, and applications to control and stochastic problems.

Abstract

We prove that the resolvent of a linear operator pencil is analytic on an open annulus if and only if the coefficients of the Laurent series satisfy a system of fundamental equations and are geometrically bounded. Our analysis extends earlier work on the fundamental equations to include the case where the resolvent has an isolated essential singularity. We find a closed form for the resolvent and use the fundamental equations to establish key spectral separation properties when the resolvent has only a finite number of isolated singularities. Finally we show that our results can also be applied to polynomial pencils.
Paper Structure (16 sections, 4 theorems, 63 equations)

This paper contains 16 sections, 4 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Motivation: Mean first-passage times for perturbed Markov processes
  3. Motivation: Input retrieval in linear control systems
  4. Previous work on spectral theory for linear operator pencils
  5. Previous work on the fundamental equations
  6. Other relevant work
  7. The main results
  8. Isolated singularities
  9. Jordan chains for a finite-order pole
  10. Jordan chains for an isolated essential singularity
  11. Solving the fundamental equations
  12. Unbounded operators
  13. Global structure of the resolvent
  14. Polynomial pencils
  15. Analytic pencils
  16. ...and 1 more sections

Key Result

Theorem 1

Let $s,r \in {\mathbb R}$ with $0 \leq s < r$. There exists an analytic resolvent $R:{\mathcal{U}}_{s,r} \rightarrow {\mathcal{L}}(K,H)$ for $A$ defined by $R(z) = \sum_{j \in {\mathbb Z}} R_j z^j$ if and only if $\{R_j\}_{j \in {\mathbb Z}} \in {\mathcal{L}}(K,H)$ are geometrically bounded by $(gb)

Theorems & Definitions (10)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Corollary 2
  • Definition 1
  • Remark 1
  • Example 1
  • Example 2: Gradient approximation
  • Example 3
  • Definition 2