The submanifolds of compact operators with fixed Jordan cells

The manifold of symmetric real matrices with fixed multiplicities of eigenvalues was first considered by V. I. Arnold. In the case of compact real self-adjoint operators, his results were generalized by the group of Japanese mathematicians, D. Fujiwara, M. Tanikawa, and Sh. Yukita. They introduced a special local diffeomorphism that maps Arnold’s submanifold to a flat subspace. Ya. Dymarskii developed the aforementioned works into a full theory. Here, we will describe the smooth structure of a submanifold of the compact operators of the general form such that the selected eigenvalue corresponds to a fixed Jordan normal form. The research is based on a straightening diffeomorphism and Arnold’s results about families of matrices depending on parameters.