%0 Journal Article
%T The abstract Titchmarsh-Weyl M-function for adjoint operator pairs and its relation to the spectrum
%A Malcolm Brown
%A James Hinchcliffe
%A Marco Marletta
%A Serguei Naboko
%A Ian Wood
%J Mathematics
%D 2008
%I arXiv
%X In the setting of adjoint pairs of operators we consider the question: 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 obtain results showing that it is possible to describe explicitly certain spaces $\Sc$ and $\tilde{\Sc}$ such that the resolvent bordered by projections onto these subspaces is analytic everywhere that the M-function is analytic. We present three examples -- one involving a Hain-L\"{u}st 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.
%U http://arxiv.org/abs/0808.3733v2