|
The index of triangular operator matrices
Author(s):
K.-H.
Förster;
B.
Nagy
Journal:
Proc. Amer. Math. Soc.
128
(2000),
1167-1176.
MSC (1991):
Primary 47A10, 47B65;
Secondary 15A18, 15A48
Posted:
August 5, 1999
MathSciNet review:
1664366
Retrieve article in:
PDF
This article is available free of charge
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.
References:
- [AB]
- C.Aliprantis and O.Burkinshaw, Positive operators, Academic Press, Orlando, 1985. MR 87h:47086
- [BCN]
- R.Bru, J.J.Climent and M.Neumann, On the index of block upper triangular matrices, SIAM J. Matrix Anal. Appl., 16 (1995), 436-447. MR 96i:15011
- [FH]
- S.Friedland and D.Hershkowitz, The rank of powers of matrices in a block triangular form, Linear Alg. Appl., 107 (1988), 17-22. MR 89h:15003
- [H]
- G. Hengstberger, Spectral properties of operator matrices, Tübinger Berichte zur Funktionalanalysis, 1 (1991/92), 65-72.
- [HRS]
- D. Hershkowitz, U.G.Rothblum and H.Schneider, The combinatorial structure of the generalized nullspace of a block triangular matrix, Linear Alg. Appl., 116 (1989), 9-26. MR 90a:15026
- [JLV]
- R-J. Jang-Lewis and H.D.Victory,Jr., On the ideal structure of positive, eventually compact linear operators on Banach lattices, Pacific J. Math., 157 (1993), 57-85. MR 93m:47042
- [MR]
- C. D. Meyer and N.J.Rose, The index and the Drazin inverse of block triangular matrices, SIAM J. Appl. Math., 33 (1977), 1-7. MR 57:345
- [N]
- R.Nagel, Towards a ``matrix theory" for unbounded operator matrices, Math. Zeitschr., 201 (1989), 57-68. MR 90c:47004
- [R]
- U.G.Rothblum, Algebraic eigenspaces of nonnegative matrices, Linear Alg. Appl., 12 (1975), 281-292. MR 53:8100
- [SBL]
- H.H.Schaefer, Banach lattices and positive operators, Springer, Berlin, 1974. MR 54:11023
- [ST]
- H.H.Schaefer, Topological vector spaces, 3rd print, Springer, Berlin, 1971. MR 49:7722
- [T]
- A.E.Taylor, Introduction to functional analysis, Wiley, New York, 1958. MR 20:5411
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical
Society
with
MSC (1991):
47A10, 47B65,
15A18, 15A48
Retrieve articles in all Journals with
MSC (1991):
47A10, 47B65,
15A18, 15A48
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:
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
Posted:
August 5, 1999
Additional Notes:
The second author was supported by the Hungarian NSF Grant (OTKA No. T-016925).
Communicated by:
David R. Larson
Copyright of article:
Copyright
2000,
American Mathematical Society
|
