CHARACTERIZATIONS OF GENERALIZED POLES BY POLE CANCELLATION FUNCTIONS OF HIGHER ORDER. MUHAMED BOROGOVAC AND ANNEMARIE LUGER http://arxiv.org/abs/1309.3677. Abstract.
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MUHAMED BOROGOVAC AND ANNEMARIE LUGER
is introduced and it is proven that for each such pole cancelation function of a given order there exists a corresponding Jordan chain of the representing linear relation of length . The converse statement is first proven under the condition that the general pole is not a zero of in the same time. For a given Jordan chain of at eigenvalue it is proven that the function
satisfies all requirements of ourdefinition of pole cancelation functions of order .
Definition 2.1 A function : belongs to generalized Nevanlinna class if
has κ negative squares i.e. for arbitrary and the Hermitian matrix has at most negative squares, and is minimal with this property.
The following representation is the main tool in the research of the generalized Nevanlinna functions.
where is a self-adjoint linear relation in a Pontryagin space , , is bounded operator.
and assume that is not a generalized zero of Q. If is a generalized pole of Q, that is , and , is a Jordan chain of A at , then
is a strong pole cancelation function of at of order .
Corollary 3.6 Under assmption of Th. 3.5
where and =, see [LaLu].
where and is an -valued polynomial of degree .