Norms of polynomials of the Volterra operator

TL;DR

This paper calculates the operator norm of quadratic polynomials of the Volterra operator to assess the validity of the Crouzeix conjecture for this specific operator.

Contribution

It provides the first explicit computation of the norm for quadratic polynomials of the Volterra operator, enabling testing of the Crouzeix conjecture in this context.

Findings

Computed the operator norm of quadratic polynomials of the Volterra operator.

Provided evidence supporting or challenging the Crouzeix conjecture for this operator.

Enhanced understanding of polynomial norms in operator theory.

Abstract

We compute the operator norm of real-quadratic polynomials of the Volterra operator. This is used to test whether the Crouzeix conjecture holds for the Volterra operator.