In Topics in matrix analysis by R. A. Horn, C. J. Johnson it's shown that for two positive matrices $A$ and $B$, $\rho(A\circ B)<\rho(A)\rho(B)$. I'm wondering whether the extension of this to positive linear operators is an established result. And is there any restrictions I should worry about?
For instance, $T_1(f)=\int k_1(x,y)f(y)dy$, $T_2(f)=\int k_2(x,y)f(y)dy$ for some strictly positive, square integrable kernels $k_1$ and $k_2$, do we naturally have $\rho(T_1\circ T_2)<\rho(T_1)\rho(T_2)$ where $T_1\circ T_2(f)=\int k_1(x,y)k_2(x,y)f(y)dy$