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Maximum principles for matrix-valued regular functions of a quaternionic variable

Sachindranath Jayaraman, Dhashna T. Pillai

Abstract

A quaternionic matrix-valued regular function is a map $F: Ω\rightarrow M_n(\mathbb{H})$ whose entries are regular functions of a quaternion variable, where $Ω$ is a domain in $\mathbb{H}$. Our aim is to bring out some maximum norm principles for such functions. We derive a decomposition theorem for such functions and also prove a Caratheodory-Rudin type approximation theorem for functions in the quaternionic right Schur class. This in turn yields that a $2 \times 2$ norm one matrix-valued function can be approximated by quaternionic rational inner functions.

Maximum principles for matrix-valued regular functions of a quaternionic variable

Abstract

A quaternionic matrix-valued regular function is a map whose entries are regular functions of a quaternion variable, where is a domain in . Our aim is to bring out some maximum norm principles for such functions. We derive a decomposition theorem for such functions and also prove a Caratheodory-Rudin type approximation theorem for functions in the quaternionic right Schur class. This in turn yields that a norm one matrix-valued function can be approximated by quaternionic rational inner functions.

Paper Structure

This paper contains 6 sections, 12 theorems, 37 equations.

Table of Contents

  1. A brief introduction
  2. Definitions and Preliminaries
  3. Main Results
  4. The Frobenius and Operator norm case
  5. Maximum principles for singular values
  6. Approximation of a quaternionic matrix-valued function

Key Result

Lemma 2.3

Let $f$ be a regular function defined on a domain $\Omega$. Then for any $I \in \mathbb{S}$ and any $J \in \mathbb{S}$ with $I \perp J$, there exists two holomorphic functions $F, G:\Omega_I \rightarrow L_I$ such that for every $z = x + yI$,

Theorems & Definitions (41)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3: Splitting lemma
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6: Identity principle
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 31 more