# Spectral sets for numerical range

## TL;DR

This paper introduces a new concept called spectral sets for the numerical range, providing criteria and dilation theorems, with a focus on the role of the base point and illustrative examples.

## Contribution

It defines a numerical-range analogue of spectral sets, establishing positivity criteria and dilation theorems, highlighting the role of the base point in the new framework.

## Key findings

- Established a positivity criterion for the new spectral set analogue
- Proved a dilation theorem similar to classical spectral set results
- Provided examples illustrating the role of the base point in the new definition

## Abstract

We define and study a numerical-range analogue of the notion of spectral set. Among the results obtained are a positivity criterion and a dilation theorem, analogous to those already known for spectral sets. An important difference from the classical definition is the role played in the new definition by the base point. We present some examples to illustrate this aspect.