# On a problem by Hans Feichtinger

## TL;DR

This paper addresses a spectral problem related to positive semi-definite trace-class pseudodifferential operators on modulation spaces, providing a counterexample that resolves the question posed by Feichtinger and extended by Heil and Larson.

## Contribution

It constructs a counterexample that solves a spectral problem on modulation spaces, extending the problem to a broader Hilbert space setting.

## Key findings

- Constructed a counterexample for the spectral problem
- Extended the problem to positive semi-definite trace-class operators on Hilbert spaces
- Provided a solution that addresses Feichtinger's original question

## Abstract

In this paper, we solve a spectral problem about positive semi-definite trace-class pseudodifferential operators on modulation spaces which was posed by H. Feichtinger. Later, C. Heil and D. Larson rephrased the problem in the broader setting of positive semi-definite trace-class operators on a separable Hilbert space. Our solution consists in constructing a counterexample that solves Hans Feichtinger's problem by first solving this second problem.