# On the Operator Equations $A^n=A^*A$

## TL;DR

This paper investigates operator equations of the form $A^*A=A^n$ for $n
eq 1$, characterizing when solutions are self-adjoint or normal, and introduces a new class of operators for $n extgreater 2$.

## Contribution

It provides a detailed analysis of the operator equations $A^*A=A^n$ for $n
eq 1$, identifying conditions for self-adjointness and normality, and introduces a new class of operators for $n extgreater 2$.

## Key findings

- Operators satisfying $A^*A=A^n$ with $n
eq 1$ are characterized.
- For $n
eq 1$, solutions are often self-adjoint or normal under certain conditions.
- A new class of operators emerges for $n extgreater 2$, positioned between orthogonal projections and normal operators.

## Abstract

Let $n\in\mathbb{N}$ and let $A$ be a closed linear operator (everywhere bounded or unbounded). In this paper, we study (among others) equations of the type $A^*A=A^n$ where $n\geq2$ and see when they yield $A=A^*$ (or a weaker class of operators). In case $n\geq3$, we have in fact a new class of operators which could placed right after orthogonal projections and just before normal operators.