  We derive sufficient conditions for strict dissipativity for optimal control
of linear evolution equations on Hilbert spaces with a cost functional
including linear and quadratic terms. We show that strict dissipativity with a
particular storage function is equivalent to ellipticity of a Lyapunov-like
operator. Further we prove under a spectral decomposition assumption of the
underlying generator and an orthogonality condition of the resulting subspaces
that this ellipticity property holds under a detectability assumption. We
illustrate our result by means of an example involving a heat equation on a
one-dimensional domain.
