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

TL;DR

This paper investigates the relationship between the Weyl M-function and the spectrum of adjoint operator pairs, providing explicit descriptions of certain subspaces where the resolvent and M-function share singularities, supported by three illustrative examples.

Contribution

It explicitly characterizes subspaces where the resolvent's analyticity matches that of the M-function in the context of adjoint operator pairs.

Findings

The M-function's singularities align with the resolvent on specific subspaces.

Explicit descriptions of these subspaces are provided.

Three diverse examples demonstrate the optimality of the results.

Abstract

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.