Unconditional convergence of eigenfunction expansions for abstract and elliptic operators

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.