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Journal of the London Mathematical Society Advance Access originally published online on March 20, 2008
Journal of the London Mathematical Society 2008 77(3):700-718; doi:10.1112/jlms/jdn006
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© 2008 London Mathematical Society

Boundary triplets and M-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices

Malcolm Brown

School of Computer Science
Cardiff University
Queen's Buildings
5 The Parade
Cardiff
CF24 3AA
United Kingdom
Malcolm.Brown@cs.cardiff.ac.uk

Marco Marletta

School of Mathematics
Cardiff University
Senghennydd Road
Cardiff
CF24 4AG
United Kingdom
MarlettaM@cardiff.ac.uk

Serguei Naboko

Department of Mathematical Physics
Institute of Physics
St. Petersburg State University
1 Ulianovskaia
St Petergoff
St Petersburg 198504
Russia
naboko@snoopy.phys.spbu.ru

Ian Wood

Institute of Mathematical and Physical Sciences
Aberystwyth University
Penglais
Aberystwyth
Ceredigion
SY 23 3BZ
United Kingdom

Starting with an adjoint pair of operators, under suitable abstract versions of standard PDE hypotheses, we consider the Weyl M-function of extensions of the operators. The extensions are determined by abstract boundary conditions and we establish results on the relationship between the M-function as an analytic function of a spectral parameter and the spectrum of the extension. We also give an example where the M-function does not contain the whole spectral information of the resolvent, and show that the results can be applied to elliptic PDEs where the M-function corresponds to the Dirichlet to Neumann map.

2000 Mathematics Subject Classification 35J25, 35P05, 47A10, 47A11.

Serguei Naboko wishes to thank the British EPSRC for supporting his visit to Cardiff under the grant EP/C008324/1 ‘Spectral Problems on Families of Domains and Operator M-functions’. He also wishes to thank Cardiff University for hospitality during the visit. Ian Wood wishes to thank British EPSRC for support under the same grant. All authors wish to thank INTAS for financial support under INTAS Project no. 051000008-7883.

Received March 21, 2007; revised January 8, 2008; published online March 20, 2008.


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