Closed numerical ranges

TL;DR

This paper establishes a precise maximum value for the sum of a bounded complex sequence and a complex number within its supremum magnitude, using properties of a related normaloid operator.

Contribution

It introduces a novel approach linking normaloid operators to bounded sequences to determine exact supremum values.

Findings

The supremum of |x_n + λ| over |λ| ≤ M_x is exactly 2M_x.

A normaloid operator associated with the sequence is used in the proof.

The result provides a clear characterization of the maximum sum magnitude for bounded sequences.

Abstract

Let $ x=(x_{n})_{n} $ be a bounded complex sequence and let $M_{x} = \sup_n |x_n|$. By using a normaloid operator related to the sequence $ x=(x_{n})_{n} $, we prove that $$ \sup_{\lambda \in \mathbb{C}, |\lambda| \leq M_x} \sup_n |x_n+\lambda| = 2M_x.$$