Operator norm and numerical radius analogues of Cohen's inequality
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2019
- DOI:
- 10.48550/arXiv.1905.08009
- arXiv:
- arXiv:1905.08009
- Bibcode:
- 2019arXiv190508009D
- Keywords:
-
- Mathematics - Functional Analysis;
- 47A30;
- 47A12;
- 47A10
- E-Print:
- 5 pages