Bounds on the joint and generalized spectral radius of Hadamard geometric mean of bounded sets of positive kernel operators

Let ${\mathrm{\Psi }}_1,\dots {\mathrm{\Psi }}_m$ be bounded sets of positive kernel operators on a Banach function space $L$. We prove that for the generalized spectral radius $\rho$ and the joint spectral radius $\widehat{\rho }$ the inequalities $$\rho ({\mathrm{\Psi }}_1^{\left(\frac{1}{m}\right)}\circ \cdots \circ {\mathrm{\Psi }}_m^{\left(\frac{1}{m}\right)})\le \rho ({\mathrm{\Psi }}_1{\mathrm{\Psi }}_2\cdots {\mathrm{\Psi }}_m{)}^{\frac{1}{m}},$$ $$\widehat{\rho }({\mathrm{\Psi }}_1^{\left(\frac{1}{m}\right)}\circ \cdots \circ {\mathrm{\Psi }}_m^{\left(\frac{1}{m}\right)})\le \widehat{\rho }({\mathrm{\Psi }}_1{\mathrm{\Psi }}_2\cdots {\mathrm{\Psi }}_m{)}^{\frac{1}{m}}$$ hold, where $\Psi _1 ^{\left( \frac{1}{m} \right)} \circ \cdots \circ \Psi _m ^{\left( \frac{1}{m} \right)}$denotestheHadamard(Schur)geometricmeanofthesets$\Psi _1, \ldots , \Psi _m$.