The Abstract Titchmarsh-Weyl M-function for Adjoint Operator Pairs and its Relation to the Spectrum

摘要

In the setting of adjoint pairs of operators we consider the ques_tion: to what extent does the Weyl M-function see the same singularities as the resolvent of a certain restriction A_B of the maximal operator? We ob_tain results showing that it is possible to describe explicitly certain spaces S and S such that the resolvent bordered by projections onto these subspaces is analytic everywhere that the M-function is analytic. We present three ex_amples - one involving a Hain-Lust type operator, one involving a perturbed Friedrichs operator and one involving a simple ordinary differential operators on a half line - which together indicate that the abstract results are probably best possible.