  We study the most general class of eigenfunction expansions for abstract
normal operators with pure point spectrum in a complex Hilbert space. We find
sufficient conditions for such expansions to be unconditionally convergent in
spaces with two norms and also estimate the degree of this convergence. Our
result essentially generalizes and complements the known theorems of M. Krein
and of Krasnosel'ski\u{\i} and Pustyl'nik. We apply it to normal elliptic
pseudodifferential operators on compact boundaryless $C^{\infty}$-manifolds. We
find generic conditions for eigenfunction expansions induced by such operators
to converge unconditionally in the Sobolev spaces $W^{\ell}_{p}$ with $p>2$ or
in the spaces $C^{\ell}$ (specifically, for the $p$-th mean or uniform
convergence on the manifold). These conditions are sufficient and necessary for
the indicated convergence on Sobolev or H\"ormander function classes and are
given in terms of parameters characterizing these classes. We also find
estimates for the degree of the convergence on such function classes. These
results are new even for differential operators on the circle and for multiple
Fourier series.
