TY  - JOUR
A1  - Schulze, Bert-Wolfgang
A1  - Seiler, Jörg
T1  - Elliptic complexes on manifolds with boundary
JF  - The journal of geometric analysis
N2  - We show that elliptic complexes of (pseudo) differential operators on smooth compact manifolds with boundary can always be complemented to a Fredholm problem by boundary conditions involving global pseudodifferential projections on the boundary (similarly as the spectral boundary conditions of Atiyah, Patodi, and Singer for a single operator). We prove that boundary conditions without projections can be chosen if, and only if, the topological Atiyah-Bott obstruction vanishes. These results make use of a Fredholm theory for complexes of operators in algebras of generalized pseudodifferential operators of Toeplitz type which we also develop in the present paper.
KW  - Elliptic complexes
KW  - Manifolds with boundary
KW  - Atiyah-Bott obstruction
KW  - Toeplitz-type pseudodifferential operators
Y1  - 2018
U6  - https://doi.org/10.1007/s12220-018-0014-6
SN  - 1050-6926
SN  - 1559-002X
VL  - 29
IS  - 1
SP  - 656
EP  - 706
PB  - Springer
CY  - New York
ER  - 
TY  - JOUR
A1  - Wallenta, Daniel
T1  - A Lefschetz fixed point formula for elliptic quasicomplexes
JF  - Integral equations and operator theor
N2  - In a recent paper, the Lefschetz number for endomorphisms (modulo trace class operators) of sequences of trace class curvature was introduced. We show that this is a well defined, canonical extension of the classical Lefschetz number and establish the homotopy invariance of this number. Moreover, we apply the results to show that the Lefschetz fixed point formula holds for geometric quasiendomorphisms of elliptic quasicomplexes.
KW  - Elliptic complexes
KW  - Fredholm complexes
KW  - Lefschetz number
Y1  - 2014
U6  - https://doi.org/10.1007/s00020-014-2122-4
SN  - 0378-620X
SN  - 1420-8989
VL  - 78
IS  - 4
SP  - 577
EP  - 587
PB  - Springer
CY  - Basel
ER  - 
TY  - JOUR
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - The dirichlet to Neumann operator for elliptic complexes
JF  - Transactions of the American Mathematical Society
N2  - We define the Dirichlet to Neumann operator for an elliptic complex of first order differential operators on a compact Riemannian manifold with boundary. Under reasonable conditions the Betti numbers of the complex prove to be completely determined by the Dirichlet to Neumann operator on the boundary.
KW  - Elliptic complexes
KW  - Dirichlet to Neumann operator
KW  - inverse problems
Y1  - 2011
SN  - 0002-9947
VL  - 363
IS  - 12
SP  - 6421
EP  - 6437
PB  - American Mathematical Soc.
CY  - Providence
ER  - 
TY  - JOUR
A1  - Wallenta, D.
T1  - Elliptic quasicomplexes on compact closed manifolds
JF  - Integral equations and operator theor
N2  - We consider quasicomplexes of pseudodifferential operators on a smooth compact manifold without boundary. To each quasicomplex we associate a complex of symbols. The quasicomplex is elliptic if this symbol complex is exact away from the zero section. We prove that elliptic quasicomplexes are Fredholm. Moreover, we introduce the Euler characteristic for elliptic quasicomplexes and prove a generalisation of the Atiyah-Singer index theorem.
KW  - Elliptic complexes
KW  - Fredholm complexes
KW  - Index theory
Y1  - 2012
U6  - https://doi.org/10.1007/s00020-012-1983-7
SN  - 0378-620X
VL  - 73
IS  - 4
SP  - 517
EP  - 536
PB  - Springer
CY  - Basel
ER  -