The abstract Titchmarsh-Weyl M-function for adjoint operator pairs and its relation to the spectrum

Details The abstract Titchmarsh-Weyl M-function for adjoint operator pairs and its relation to the spectrum The abstract Titchmarsh-Weyl M-function for adjoint operator pairs and its relation to the spectrum

2008-08-27

arxiv.org/abs/0808.3733v2

Mathematics

Spectral Theory

21 pages minor amendments, extra references

Scientific paper

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.

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