# A note on Anderson's theorem in the infinite-dimensional setting

## TL;DR

This paper explores an extension of Anderson's theorem to infinite-dimensional operators, specifically focusing on the sum of a normal and a compact operator, building on prior finite-dimensional and compact operator results.

## Contribution

It extends Anderson's theorem to the case where the operator is the sum of a normal and a compact operator in an infinite-dimensional setting.

## Key findings

- Established conditions under which the numerical range coincides with the unit disk.
- Generalized finite-dimensional results to a broader class of infinite-dimensional operators.
- Provided new insights into the spectral properties of sums of normal and compact operators.

## Abstract

Anderson's theorem states that if the numerical range W(A) of an n-by-n matrix A is contained in the unit disk and intersects with the unit circle at more than n points, then it coincides with the (closed) unit dissk. An analogue of this result for compact A in an infinite dimensional setting was established by Gau and Wu. We consider here the case of A being the sum of a normal and compact operator.