On commuting compact self-adjoint operators on a Pontryagin space

Suppose that A(1), A(2),...A(n) are compact, commuting self-adjoint linear maps on a Pontryagin space K of index k and that their joint root subspace M-o at the zero eigenvalue in C-n is a nondegenerate subspace. Then there exist joint invariant subspaces H and F in K such that K = F + H, H is a Hilbert space and F is finite-dimensional space with k less than or equal to dim F less than or equal to (n + 2)k. We also consider the structure of restrictions A(j)\F in the case k = 1.