  In this work, we prove a version of H\"{o}rmander's theorem for a stochastic
evolution equation driven by a trace-class fractional Brownian motion with
Hurst exponent $\frac{1}{2} < H < 1$ and an analytic semigroup on a given
separable Hilbert space. In contrast to the classical finite-dimensional case,
the Jacobian operator in typical solutions of parabolic stochastic PDEs is not
invertible which causes a severe difficulty in expressing the Malliavin matrix
in terms of an adapted process. Under a H\"{o}rmander's bracket condition and
some algebraic constraints on the vector fields combined with the range of the
semigroup, we prove the law of finite-dimensional projections of such solutions
has a density w.r.t Lebesgue measure. The argument is based on rough path
techniques and a suitable analysis on the Gaussian space of the fractional
Brownian motion.

Probability