In this paper we analyse the structure of the spaces of coefficients of
eigenfunction expansions of functions in Komatsu classes on compact manifolds,
continuing the research in our previous paper. We prove that such spaces of
Fourier coefficients are perfect sequence spaces. As a consequence we describe
the tensor structure of sequential mappings on spaces of Fourier coefficients
and characterise their adjoint mappings. In particular, the considered classes
include spaces of analytic and Gevrey functions, as well as spaces of
ultradistributions, yielding tensor representations for linear mappings between
these spaces on compact manifolds.