  We provide an abstract framework for a Logvinenko-Sereda type theorem, where
the classical compactness assumption on the support of the Fourier transform is
replaced by the assumption that the functions under consideration belong to a
spectral subspace associated with a finite energy interval for some lower
semibounded self-adjoint operator on a Euclidean $L^2$-space. Our result then
provides a bound for the $L^2$-norm of such functions in terms of their
$L^2$-norm on a thick subset with a constant explicit in the geometric and
spectral parameters. This recovers previous results for functions on the whole
space, hyperrectangles, and infinite strips with compact Fourier support and
for finite linear combinations of Hermite functions and allows to extend them
to other domains.
  The proof follows the approach by Kovrijkine and is based on Bernstein-type
inequalities for the respective functions, complemented with a suitable
covering of the underlying domain.
