Let $\Psi$ and $\Sigma$ be bounded sets of positive kernel operators on a Banach function space $L$. We prove several refinements of the known inequalities $$\rho \left(\Psi ^{ \left( \frac{ 1}{ 2} \right)} \circ \Sigma ^{ \left( \frac{ 1}{ 2} \right)} \right) \le \rho (\Psi \Sigma) ^{ \frac{ 1}{ 2}} \;\; \mathrm{ and}\;\; \hat{ \rho} \left(\Psi ^{ \left( \frac{ 1}{ 2} \right)} \circ \Sigma ^{ \left( \frac{ 1}{ 2} \right)} \right) \le \hat{ \rho} (\Psi \Sigma) ^{ \frac{ 1}{ 2}} $$ for the generalized spectral radius $\rho$ and the joint spectral radius $\hat{ \rho}$, where $\Psi ^{ \left( \frac{ 1}{ 2} \right)} \circ \Sigma ^{ \left( \frac{ 1}{ 2} \right)} $ denotes the Hadamard (Schur) geometric mean of the sets $\Psi $ and $\Sigma$. Furthermore, we prove that analogous inequalities hold also for the generalized essential spectral radius and the joint essential spectral radius in the case when $L$ and its Banach dual $L^*$ have order continuous norms.
Categories: math.FA