  We propose notions of minimax and viscosity solutions for a class of fully
nonlinear path-dependent PDEs with nonlinear, monotone, and coercive operators
on Hilbert space. Our main result is well-posedness (existence, uniqueness, and
stability) for minimax solutions. A particular novelty is a suitable
combination of minimax and viscosity solution techniques in the proof of the
comparison principle. One of the main difficulties, the lack of compactness in
infinite-dimensional Hilbert spaces, is circumvented by working with suitable
compact subsets of our path space. As an application, our theory makes it
possible to employ the dynamic programming approach to study optimal control
problems for a fairly general class of (delay) evolution equations in the
variational framework. Furthermore, differential games associated to such
evolution equations can be investigated following the Krasovskii-Subbotin
approach similarly as in finite dimensions.
