  We analyze strict dissipativity of generalized linear quadratic optimal
control problems on Hilbert spaces. Here, the term ``generalized'' refers to
cost functions containing both quadratic and linear terms. We characterize
strict pre-dissipativity with a quadratic storage function via coercivity of a
particular Lyapunov-like quadratic form. Further, we show that under an
additional algebraic assumption, strict pre-dissipativity can be strengthened
to strict dissipativity. Last, we relate the obtained characterizations of
dissipativity with exponential detectability.
