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

DOI:
https://doi.org/10.1090/S0002-9939-99-05341-1

Published electronically:
August 5, 1999

MathSciNet review:
1664366

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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:
https://doi.org/10.1090/S0002-9939-99-05341-1

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