Analytic characterizations of Jordan chains by pole cancellation functions of higher order
Scalar Nevanlinna functions are functions holomorphic on \(\mathbb C \setminus \mathbb R\), which are symmetric with respect to the real line and map the upper half plane into itself. There is an obvious generalization to vector valued functions. Obviously, the resolvent of a self-adjoint operator in a Hilbert space is an example for such a function. Vice versa, each Nevanlinna function admits a representation with the help of a self-adjoint operator (or relation) in a Hilbert space.NEWLINENEWLINESince many years generalized Nevanlinna functions are studied. This is a generalization of the above mentioned Nevanlinna functions. In particular, generalized Nevanlinna functions admit a representation with a self-adjoint operator (or relation) in a Pontryagin space. It is well known that self-adjoint operators (or relations) in Pontryagin spaces may have eigenvalues with Jordan chains of length greater than one. In the scalar case it is known how to characterize these Jordan chains in analytic terms, i.e., in terms of the generalized Nevanlinna function and its derivatives. In the non-scalar case this question was open and only partially results had been obtained so far. In this paper this gap is closed. It is shown that the Jordan chains can be fully constructed via so-called pole cancelation functions, including the signs (with respect to the indefinite inner product of the Pontryagin space) of the elements. Moreover, and this is the novelty and the main contribution, it is shown that the pole cancelation function is always of a surprisingly simple form involving only a polynomial, a constant operator and the generalized Nevanlinna function. In particular, under some additional conditions (like regularity), the constant operator can be chosen to be zero and the polynomial has the elements of the Jordan chain as coefficients. Hence, the pole cancelation function can be determinant in a very easy and concrete way which makes it a more accessible tool in spectral theory.
scientific article; zbMATH DE number 6361468
scientific article