# Equivalence between estimates for quasi-commutators and commutators

## TL;DR

This paper demonstrates that under certain conditions, estimates for quasi-commutators can be derived from those for commutators, simplifying the analysis of operator estimates in functional analysis.

## Contribution

It establishes an equivalence between estimates for quasi-commutators and commutators under mild conditions, reducing the complexity of such estimates.

## Key findings

- Estimates for quasi-commutators can be obtained from commutator estimates.
- The equivalence holds under mild conditions on the space and operators.
- Simplifies the analysis of operator estimates in functional analysis.

## Abstract

In this short note we show that under some mild conditions on the space and the operators, an estimate for $\|Sf(A) - f(B)S\|$ needs only to be studied for invertible $S$ and $B$ equal to $A$. Thus estimates for a quasi-commutator can be derived from that for the commutator.