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

• Published in: Integral equations and operator theory Vol. 63; no. 3; pp. 297 - 320
• Main Authors: Brown, Malcolm, Hinchcliffe, James, Marletta, Marco, Naboko, Serguei, Wood, Ian
• Format: Journal Article
• Language: English
• Published: Basel Birkhäuser-Verlag 01-03-2009
• Subjects: Analysis; Mathematics; Mathematics and Statistics; Secondary 35J25, 35P05, 47A11; adjoint pairs; boundary triples; spectrum; Primary 47A10; function
• ISSN: 0378-620X; 1420-8989
• DOI: 10.1007/s00020-009-1668-z

.
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
and
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.