Representation of invariant subspaces of the Schwartz space

A subspace $W$ of the Schwartz space $C^{\infty} (a,b)$ such that the restriction of the operator of differentiation to $W$ has a discrete spectrum is considered. Conditions for the representation of $W$ as a direct algebraic and topological sum of two subspaces, namely, the residual subspace and the subspace spanned by the exponential monomials from $W$, are investigated. One condition ensuring this representation turns out to be the existence of a functional annihilating $W$ such that the Fourier-Laplace transform of this functional is a slowly decreasing entire function. A new characteristic of complex sequences is introduced and investigated. Using this characteristic, the condition that an invariant subspace is equal to the direct sum of its residual and exponential subspaces can be put into a form that is similar to the previously discovered conditions for the possibility of weak spectral synthesis.Bibliography: 19 titles.

invariant subspace, spectral synthesis, Fourier-Laplace transform, slowly decreasing entire function, Schwartz spaces