  In 1992, Szyld provided a sequence of lower bounds for the spectral radius of
a nonnegative matrix $A$, based on the geometric symmetrization of powers of
$A$. In 1998, Ta\c{s}\c{c}i and Kirkland proved a companion result by giving a
sequence of upper bounds for the spectral radius of $A$, based on the
arithmetic symmetrization of powers of $A$. In this note, we extend both
results to positive operators on $L^2$-spaces.
