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

2009; Birkhäuser; Volume: 63; Issue: 3 Linguagem: Inglês

10.1007/s00020-009-1668-z

1420-8989

Malcolm Brown, James Hinchcliffe, Marco Marletta, Serguei Naboko, Ian Wood,

Differential Equations and Boundary Problems

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 $${\mathcal{S}}$$ and $$\tilde{\mathcal{S}}$$ 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-Lüst 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.