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With the aim of applications to solving general integral equations, weintroduce 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, whoseproperty of being infinitely smooth is stable under passage to certain left andright multiples of their associated integral operators. An expansion theorem inabsolutely and uniformly convergent bilinear series concerning kernels of thisclass is proved extending to a general non-Hermitian setting both Mercer's andKadota's Expansion Theorems for positive definite kernels. Another theoremproved in this paper identifies families of those bounded operators on aseparable Hilbert space $\mathcal{H}$ that can be simultaneously transformed bythe same unitary equivalence transformation into bi-Carleman integral operatorson $L^2(\mathbb{R})$, whose kernels are $K^\infty$ kernels of Mercer type; itssingleton version implies in particular that any bi-integral operator isunitarily equivalent to an integral operator with such a kernel.

Integral operators with infinitely smooth bi-Carleman kernels of Mercer type