Some new relations between the Berezin number and the Berezin norm of operators

Abstract

We prove new Grüss type inequalities for the Berezin symbol of some operator products. As biproducts, we find new relations between the Berezin number, the Berezin norm and some new quantities for operators acting on the reproducing kernel Hilbert space. In particular, we prove for arbitrary bounded linear operator $A$ that

$$\text{ber}(A^2)\le \frac{1}{4}\left((\text{ber}(A^4))+\parallel A^2{\parallel }_{\text{Ber}}^2\right),$$

where $\text{ber}(\cdot )$ and $\parallel \cdot {\parallel }_{\text{Ber}}$ denote, respectively, the Berezin number and the Berezin norm of operator $A$.