Some analytic properties of the Weyl function of a closed linear relation

  • O.G. Storozh Ivan Franko Lviv National University, 1 Universytetska str., 79000, Lviv, Ukraine
Keywords: Hilbert space, relation, operator, extension, pole
Published online: 2018-07-03

Abstract


Let $L$ and $L_{0}$, where $L$ is an expansion of $L_{0}$, be closed linear relations (multivalued operators) in a Hilbert space $H$. In terms of abstract boundary operators (i.e. in the form which in the case of differential operators leads immediately to boundary conditions) some analytic properties of the Weyl function $M(\lambda)$ corresponding to a certain boundary pair of the couple $(L, L_{0}),$ are studied.

In particular, applying Hilbert resolvent identity for relations, the criterion of invertibility in the algebra of bounded linear operators in $H$ for transformation $M(\lambda) - M(\lambda_{0})$ in certain small punctured neighbourhood of $\lambda_{0} $ is established. It is proved that in this case $\lambda _{0}$ is a first-order pole for the operator-function $\left(M(\lambda )- M(\lambda_{0} )\right)^{-1} $. The corresponding residue and Laurent series expansion are found.

Under some additional assumptions, the behaviour of so called $\gamma$-field $Z_{\lambda}$ (being an operator-function closely connected to $M(\lambda)$) as $\lambda \to - \infty $ is investigated.

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How to Cite
(1)
Storozh O. Some Analytic Properties of the Weyl Function of a Closed Linear Relation. Carpathian Math. Publ. 2018, 10 (1), 197-205.