Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the invariant faces associated with a cone-preserving map


Authors: Bit-Shun Tam and Hans Schneider
Journal: Trans. Amer. Math. Soc. 353 (2001), 209-245
MSC (2000): Primary 15A48; Secondary 47B65, 47A25, 46B42
DOI: https://doi.org/10.1090/S0002-9947-00-02597-6
Published electronically: July 12, 2000
MathSciNet review: 1707205
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

For an $n\times n$ nonnegative matrix $P$, an isomorphism is obtained between the lattice of initial subsets (of $\{ 1,\cdots,n\}$) for $P$ and the lattice of $P$-invariant faces of the nonnegative orthant $\mathbb{R}^{n}_{+}$. Motivated by this isomorphism, we generalize some of the known combinatorial spectral results on a nonnegative matrix that are given in terms of its classes to results for a cone-preserving map on a polyhedral cone, formulated in terms of its invariant faces. In particular, we obtain the following extension of the famous Rothblum index theorem for a nonnegative matrix: If $A$ leaves invariant a polyhedral cone $K$, then for each distinguished eigenvalue $\lambda$ of $A$ for $K$, there is a chain of $m_\lambda$ distinct $A$-invariant join-irreducible faces of $K$, each containing in its relative interior a generalized eigenvector of $A$corresponding to $\lambda$ (referred to as semi-distinguished $A$-invariant faces associated with $\lambda$), where $m_\lambda$ is the maximal order of distinguished generalized eigenvectors of $A$ corresponding to $\lambda$, but there is no such chain with more than $m_\lambda$ members. We introduce the important new concepts of semi-distinguished $A$-invariant faces, and of spectral pairs of faces associated with a cone-preserving map, and obtain several properties of a cone-preserving map that mostly involve these two concepts, when the underlying cone is polyhedral, perfect, or strictly convex and/or smooth, or is the cone of all real polynomials of degree not exceeding $n$ that are nonnegative on a closed interval. Plentiful illustrative examples are provided. Some open problems are posed at the end.


References [Enhancements On Off] (What's this?)

  • [Bar1] Barker, G. P., Perfect cones. Linear Algebra Appl. 22, 211-221 (1978) MR 80a:15020
  • [Bar2] Barker, G. P., Theory of cones. Linear Algebra Appl. 39, 263-291 (1981) MR 83e:15022
  • [B-P] Berman, A., Plemmons, R. J. Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York, 1979 MR 82b:15013
  • [B-T] Barker, G. P., Thompson, A., Cones of polynomials. Portugal. Math. 44, 183-197 (1987) MR 88k:52005
  • [Bir] Birkhoff, G., Lattice Theory. 3rd ed., American Mathematical Society, Providence, R.I., 1966 MR 37:2638
  • [Dod] Dodds, P. G., Positive compact operators. Quaestiones Math. 18, 21-45 (1995) MR 96c:00020
  • [G-L-R] Gohberg, L., Lancaster, P., Rodman, L., Invariant Subspaces of Matrices with Applications. John Wiley & Sons, New York, 1986 MR 88a:15001
  • [H-S] Hershkowitz, D., Schneider, H., On the generalized nullspace of $M$-matrices and $Z$-matrices. Linear Algebra Appl. 106, 5-23 (1988) MR 90b:15016
  • [J-V1] Jang, R., Victory, H.D., Jr., On nonnegative solvability of linear integral equations. Linear Algebra Appl. 165, 197-228 (1992) MR 93c:47083
  • [J-V2] Jang-Lewis, R., Victory, H.D., Jr., On the ideal structure of positive, eventually compact linear operators on Banach lattices. Pacific J. Math. 157, 57-85 (1993) MR 93m:47042
  • [J-V3] Jang-Lewis, R., Victory, H.D., Jr., On nonnegative solvability of linear operator equations. Integr. Equat. Oper. Th. 18, 88-108 (1994) MR 94m:47075
  • [MN1] Meyer-Nieberg, P., A partial spectral reduction for positive linear operators. Arch. Math. 45, 34-41 (1985) MR 87a:47065
  • [MN2] Meyer-Nieberg, P., Banach Lattices, Springer-Verlag, New York, 1991 MR 93f:46025
  • [Nel] Nelson, Jr., P. The structure of a positive linear integral operator. J. London Math. Soc. (2) 8, 711-718 (1974) MR 50:8169
  • [Roc] Rockafellar, R. T., Convex Analysis. Princeton Univ. Press, Princeton, NJ, 1970 MR 43:445
  • [Rot] Rothblum, U. G., Algebraic eigenspaces of nonnegative matrices. Linear Algebra Appl. 12, 281-292 (1975) MR 53:8100
  • [Scha] Schaefer, H. H., Banach lattices and positive operators. Springer-Verlag, Berlin-Heidelberg-New York, 1974 MR 54:11023
  • [Schn1] Schneider, H., The elementary divisors associated with $0$ of a singular $M$-matrix. Proc. Edinburgh Math. Soc. (2) 10, 108-122 (1956) MR 17:935d
  • [Schn2] Schneider, H., Geometric conditions for the existence of positive eigenvalues of matrices. Linear Algebra Appl. 38, 253-271 (1981) MR 83f:15012
  • [Schn3] Schneider, H., The influence of the marked reduced graph of a nonnegative matrix on the Jordan form and on related properties$\,$: a survey. Linear Algebra Appl. 84, 161-189 (1986) MR 88b:15010
  • [Su-T] Sung, C.H., Tam, B. S., A study of projectionally exposed cones. Linear Algebra Appl. 139, 225-252 (1990) MR 91j:52008
  • [Tam1] Tam, B. S., A note on polyhedral cones. J. Austral. Math. Soc. Ser A 22, 456-461 (1976) MR 55:2962
  • [Tam2] Tam, B. S., On the duality operator of a convex cone. Linear Algebra Appl. 64, 33-56 (1985) MR 86j:90118
  • [Tam3] Tam, B. S. On the distinguished eigenvalues of a cone-preserving map. Linear Algebra Appl. 131, 17-37 (1990) MR 91d:15040
  • [Tam4] Tam, B. S., On semipositive bases for a cone-preserving map, in preparation.
  • [T-S1] Tam, B. S., Schneider, H., On the core of a cone-preserving map. Trans. Amer. Math. Soc. 343, 479-524 (1994) MR 94h:15011
  • [T-S2] Tam, B. S., Schneider, H., Linear equations over cones, Collatz-Wielandt numbers and alternating sequences, in preparation.
  • [T-W] Tam, B. S., Wu, S. F., On the Collatz-Wielandt sets associated with a cone-preserving map. Linear Algebra Appl. 125, 77-95 (1989) MR 90i:15021
  • [Vic1] Victory, H. D., Jr., On linear integral operators with nonnegative kernels. J. Math. Anal. Appl. 89, 420-441 (1982) MR 84j:45003
  • [Vic2] Victory, H. D., Jr., The structure of the algebraic eigenspace to the spectral radius of eventually compact, nonnegative integral operators. J. Math. Anal. Appl. 90, 484-516 (1982) MR 84d:47041
  • [Zer] Zerner, M., Quelques propriétés spectrales des opérateurs positifs. J. Funct. Anal. 72, 381-417 (1987). MR 88i:47020

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 15A48, 47B65, 47A25, 46B42

Retrieve articles in all journals with MSC (2000): 15A48, 47B65, 47A25, 46B42


Additional Information

Bit-Shun Tam
Affiliation: Department of Mathematics, Tamkang University, Tamsui, Taiwan 25137, ROC
Email: bsm01@mail.tku.edu.tw

Hans Schneider
Affiliation: Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706
Email: hans@math.wisc.edu

DOI: https://doi.org/10.1090/S0002-9947-00-02597-6
Keywords: Cone-preserving map, nonnegative matrix, polyhedral cone, perfect cone, strictly convex smooth cone, spectral pair of a vector, spectral pair of a face, Perron-Schaefer condition, initial subset, semi-distinguished class, semi-distinguished invariant face, distinguished generalized eigenvector, chain of invariant faces
Received by editor(s): October 31, 1997
Received by editor(s) in revised form: March 11, 1999
Published electronically: July 12, 2000
Additional Notes: Research of the first author partially supported by the National Science Council of the Republic of China grant NSC 86-2115-M-032-002; the second author’s research partially supported by NSF grants DMS-9123318 and DMS-9424346.
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society