# Operator norm and numerical radius analogues of Cohen's inequality

## TL;DR

This paper establishes new inequalities relating the operator norm and numerical radius of operators on $L^2$ spaces, extending Cohen's inequality through operator norm and numerical radius analogues involving invertible multiplication operators.

## Contribution

It introduces novel inequalities connecting the operator norm and numerical radius of operators with invertible multiplication operators, extending Cohen's inequality to these contexts.

## Key findings

- Operator norm inequality: A^2 \u2264 D A  D^{-1} A
- Numerical radius inequality for positive operators: w(A)^2  w(D A)  w(D^{-1} A)
- Extension of Cohen's inequality to operator norm and numerical radius contexts

## Abstract

Let $D$ be an invertible multiplication operator on $L^2(X, \mu)$, and let $A$ be a bounded operator on $L^2(X, \mu)$. In this note we prove that $\|A\|^2 \le \|D A\| \, \|D^{-1} A\|$, where $\|\cdot\|$ denotes the operator norm. If, in addition, the operators $A$ and $D$ are positive, we also have $w(A)^2 \le w(D A) \, w(D^{-1} A)$, where $w$ denotes the numerical radius.