Dirac operator on the restricted Grassmannian manifold

In his book Mickelsson notices that the infinite-dimensional Grassmannian manifold of Segal and Wilson admits a Spin^c structure and after this he naturally considers the problem of defining a Dirac operator on it. Mickelsson gives a possible candidate for such an operator but unfortunately it proves out to be badly diverging and he leaves it as an open problem to introduce proper modifications to his original construction in order to obtain a well-defined (unbounded) operator with expected properties. Using fermionic Fock space representations of the restricted unitary group associated to a polarized Hilbert space, introduced in the well-known book written by Pressley and Segal on loop groups, we construct a well-defined candidate for a Dirac type operator on the restricted Grassmannian manifold acting on a relevant space of spinors. As our main result we show that our operator is an unbounded symmetric operator with finite-dimensional kernel.