Index Theorems for Polynomial Pencils

We survey index theorems counting eigenvalues of linearized Hamiltonian systems and characteristic values of polynomial operator pencils. We present a simple common graphical interpretation and generalization of the index theory using the concept of graphical Krein signature. Furthermore, we prove that derivatives of an eigenvector u= u(λ) of an operator pencil L(λ) satisfying L(λ) u(λ)= μ(λ) u(λ) evaluated at a characteristic value of L(λ) do not only generate an arbitrary chain of root vectors of L(λ) but the chain that carries an extra information.

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