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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The index of triangular operator matrices


Authors: K.-H. Förster and B. Nagy
Journal: Proc. Amer. Math. Soc. 128 (2000), 1167-1176
MSC (1991): Primary 47A10, 47B65; Secondary 15A18, 15A48
Published electronically: August 5, 1999
MathSciNet review: 1664366
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Abstract: For any triangular operator matrix acting in a direct sum of complex Banach spaces, the order of a pole of the resolvent (i.e. the index) is determined as a function of the coefficients in the Laurent series for all the (resolvents of the) operators on the diagonal and of the operators below the diagonal. This result is then applied to the case of certain nonnegative operators in Banach lattices. We show how simply these results imply the Rothblum Index Theorem (1975) for nonnegative matrices. Finally, examples for calculating the index are presented.


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Additional Information

K.-H. Förster
Affiliation: Department of Mathematics, Technical University Berlin, Sekr. MA 6-4, Strasse des 17. Juni 135, D-10623 Berlin, Germany
Email: foerster@math.tu-berlin.de

B. Nagy
Affiliation: Department of Analysis, Institute of Mathematics, Technical University Budapest, H-1521 Budapest, Hungary
Email: bnagy@math.bme.hu

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05341-1
PII: S 0002-9939(99)05341-1
Keywords: Triangular operator matrix, order of pole, Banach lattice, nonnegative operator and matrix, Index Theorem
Received by editor(s): October 10, 1997
Received by editor(s) in revised form: June 17, 1998
Published electronically: August 5, 1999
Additional Notes: The second author was supported by the Hungarian NSF Grant (OTKA No. T-016925).
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society