  With the aim of applications to solving general integral equations, we
introduce and study in this paper a special class of bi-Carleman kernels on
$\mathbb{R}\times\mathbb{R}$, called $K^\infty$ kernels of Mercer type, whose
property of being infinitely smooth is stable under passage to certain left and
right multiples of their associated integral operators. An expansion theorem in
absolutely and uniformly convergent bilinear series concerning kernels of this
class is proved extending to a general non-Hermitian setting both Mercer's and
Kadota's Expansion Theorems for positive definite kernels. Another theorem
proved in this paper identifies families of those bounded operators on a
separable Hilbert space $\mathcal{H}$ that can be simultaneously transformed by
the same unitary equivalence transformation into bi-Carleman integral operators
on $L^2(\mathbb{R})$, whose kernels are $K^\infty$ kernels of Mercer type; its
singleton version implies in particular that any bi-integral operator is
unitarily equivalent to an integral operator with such a kernel.
