Invariant subspaces in non-quasianalytic spaces of $\Omega$-ultradifferentiable functions on an interval

We consider and solve a weakened version of the classical spectral synthesis problem for di erentiation operator in non-quasianalytic spaces of ultradi erentiable functions (UDF). Moreover, we deal with the widest class of UDF among all known ones. Namely, we study the spaces of $\Omega$-ultradifferentiable functions introduced by Alexander Abanin in 2007–08. For subspaces of these spaces which are invariant under the di erentiation operator we establish general conditions of weak spectral synthesis.