  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.
