Inequalities on the spectral radius, operator norm and numerical radius of Hadamard weighted geometric mean of positive kernel operators

Recently, several authors have proved inequalities on the spectral radius $\rho$, operator norm $\|\cdot\|$ and numerical radius of Hadamard products and ordinary products of non-negative matrices that define operators on sequence spaces, or of Hadamard geometric mean and ordinary products of positive kernel operators on Banach function spaces. In the present article we generalize and refine several of these results. In particular, we show that for a Hadamard geometric mean $A ^{\left( \frac{1}{2} \right)} \circ B ^{\left( \frac{1}{2} \right)}$ of positive kernel operators $A$ and $B$ on a Banach function space $L$, we have $$ \rho \left(A^{(\frac{1}{2})} \circ B^{(\frac{1}{2})} \right) \le \rho \left((AB)^{(\frac{1}{2})} \circ (BA)^{(\frac{1}{2})}\right)^{\frac{1}{2}} \le \rho (AB)^{\frac{1}{2}}.$$ In the special case $L=L^2(X, \mu)$ we also prove that $$\|A^{(\frac{1}{2})} \circ B^{(\frac{1}{2})} \| \le \rho \left ( (A^* B ) ^{(\frac{1}{2})}\circ (B ^* A)^{(\frac{1}{2})}\right)^{\frac{1}{2}} \le \rho (A^* B )^{\frac{1}{2}}.$$