# The Fuglede Theorem and Some Intertwining Relations

## TL;DR

This paper extends the Fuglede Theorem to unbounded operators and explores intertwining relations, providing new versions, counterexamples, and examples of operators with unique intertwining properties.

## Contribution

It presents a new unbounded version of the Fuglede Theorem and constructs operators with unique intertwining characteristics, advancing understanding of operator relations.

## Key findings

- A new version of the Fuglede Theorem for unbounded operators
- Counterexample illustrating limitations of intertwining operators
- Example of operators not intertwined except by the zero operator

## Abstract

In this paper, we show a new and classic version of the celebrated Fuglede Theorem in an unbounded setting. A related counterexample is equally presented. In the second strand of the paper, we give a pair of a closed and self-adjoint (unbounded) operators which is not intertwined by any (bounded or closed) operator except the zero operator.