  We show, by means of a class of examples, that if $K_1$ and $K_2$ are two
positive definite kernels on the unit disc such that the multiplication by the
coordinate function on the corresponding reproducing kernel Hilbert space is
subnormal, then the multiplication operator on the Hilbert space determined by
their sum $K_1+K_2$ need not be subnormal. This settles a recent conjecture of
Gregory T. Adams, Nathan S. Feldman and Paul J. McGuire in the negative. We
also discuss some cases for which the answer is affirmative.
