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On the real and imaginary parts of powers of the Volterra operator

Thomas Ransford, Dashdondog Tsedenbayar

Abstract

We study the real and imaginary parts of the powers of the Volterra operator on $L^2[0,1]$, specifically their eigenvalues, their norms and their numerical ranges.

On the real and imaginary parts of powers of the Volterra operator

Abstract

We study the real and imaginary parts of the powers of the Volterra operator on , specifically their eigenvalues, their norms and their numerical ranges.
Paper Structure (4 sections, 16 theorems, 61 equations, 2 tables)

This paper contains 4 sections, 16 theorems, 61 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Eigenvalues
  3. Norms
  4. Numerical ranges

Key Result

Theorem 2.1

If $n$ is odd, then $\operatorname{Re} V^n$ has at most $n$ non-zero eigenvalues. If $n$ is even, then $\operatorname{Im} V^n$ has at most $n$ non-zero eigenvalues.

Theorems & Definitions (29)

  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • proof
  • Theorem 2.6
  • proof
  • Theorem 2.7
  • ...and 19 more