## On the invariant faces associated with a cone-preserving map

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- by Bit-Shun Tam and Hans Schneider
- Trans. Amer. Math. Soc.
**353**(2001), 209-245 - DOI: https://doi.org/10.1090/S0002-9947-00-02597-6
- Published electronically: July 12, 2000
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## 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

- George Phillip Barker,
*Perfect cones*, Linear Algebra Appl.**22**(1978), 211–221. MR**510810**, DOI 10.1016/0024-3795(78)90072-1 - George Phillip Barker,
*Theory of cones*, Linear Algebra Appl.**39**(1981), 263–291. MR**625256**, DOI 10.1016/0024-3795(81)90310-4 - Abraham Berman and Robert J. Plemmons,
*Nonnegative matrices in the mathematical sciences*, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR**544666** - George P. Barker and Alex Thompson,
*Cones of polynomials*, Portugal. Math.**44**(1987), no. 2, 183–197. MR**911441** - Garrett Birkhoff,
*Lattice theory*, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR**0227053** *First International Conference in Abstract Algebra*, South African Mathematical Society, Pretoria, 1995. Papers from the conference (ICAA, 1993) held in Kruger Park, April 13–23, 1993; Quaestiones Math. 18 (1995), no. 1-3. MR**1340467**- I. Gohberg, P. Lancaster, and L. Rodman,
*Invariant subspaces of matrices with applications*, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1986. A Wiley-Interscience Publication. MR**873503** - Daniel Hershkowitz and Hans Schneider,
*On the generalized nullspace of $M$-matrices and $Z$-matrices*, Linear Algebra Appl.**106**(1988), 5–23. MR**951824**, DOI 10.1016/0024-3795(88)90019-5 - Ruey-Jen Jang-Lewis and Harold Dean Victory Jr.,
*On nonnegative solvability of linear integral equations*, Linear Algebra Appl.**165**(1992), 197–228. MR**1149755**, DOI 10.1016/0024-3795(92)90238-6 - Ruey-Jen Jang-Lewis and Harold Dean Victory Jr.,
*On the ideal structure of positive, eventually compact linear operators on Banach lattices*, Pacific J. Math.**157**(1993), no. 1, 57–85. MR**1197045** - Ruey-Jen Jang-Lewis and Harold Dean Victory Jr.,
*On nonnegative solvability of linear operator equations*, Integral Equations Operator Theory**18**(1994), no. 1, 88–108. MR**1250760**, DOI 10.1007/BF01225214 - Peter Meyer-Nieberg,
*A partial spectral reduction for positive linear operators*, Arch. Math. (Basel)**45**(1985), no. 1, 34–41. MR**799447**, DOI 10.1007/BF01194004 - Peter Meyer-Nieberg,
*Banach lattices*, Universitext, Springer-Verlag, Berlin, 1991. MR**1128093**, DOI 10.1007/978-3-642-76724-1 - Paul Nelson Jr.,
*The structure of a positive linear integral operator*, J. London Math. Soc. (2)**8**(1974), 711–718. MR**355695**, DOI 10.1112/jlms/s2-8.4.711 - R. Tyrrell Rockafellar,
*Convex analysis*, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR**0274683** - Uriel G. Rothblum,
*Algebraic eigenspaces of nonnegative matrices*, Linear Algebra Appl.**12**(1975), no. 3, 281–292. MR**404298**, DOI 10.1016/0024-3795(75)90050-6 - Helmut H. Schaefer,
*Banach lattices and positive operators*, Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York-Heidelberg, 1974. MR**0423039** - Saunders MacLane,
*Steinitz field towers for modular fields*, Trans. Amer. Math. Soc.**46**(1939), 23–45. MR**17**, DOI 10.1090/S0002-9947-1939-0000017-3 - Hans Schneider,
*Geometric conditions for the existence of positive eigenvalues of matrices*, Linear Algebra Appl.**38**(1981), 253–271. MR**636041**, DOI 10.1016/0024-3795(81)90025-2 - Hans Schneider,
*The influence of the marked reduced graph of a nonnegative matrix on the Jordan form and on related properties: a survey*, Proceedings of the symposium on operator theory (Athens, 1985), 1986, pp. 161–189. MR**872282**, DOI 10.1016/0024-3795(86)90313-7 - Chen Han Sung and Bit Shun Tam,
*A study of projectionally exposed cones*, Linear Algebra Appl.**139**(1990), 225–252. MR**1071712**, DOI 10.1016/0024-3795(90)90401-W - Bit Shun Tam,
*A note on polyhedral cones*, J. Austral. Math. Soc. Ser. A**22**(1976), no. 4, 456–461. MR**429954**, DOI 10.1017/s1446788700016311 - Bit Shun Tam,
*On the duality operator of a convex cone*, Linear Algebra Appl.**64**(1985), 33–56. MR**776515**, DOI 10.1016/0024-3795(85)90265-4 - Bit Shun Tam,
*On the distinguished eigenvalues of a cone-preserving map*, Linear Algebra Appl.**131**(1990), 17–37. MR**1057062**, DOI 10.1016/0024-3795(90)90372-J - Tam, B. S., On semipositive bases for a cone-preserving map, in preparation.
- Bit Shun Tam and Hans Schneider,
*On the core of a cone-preserving map*, Trans. Amer. Math. Soc.**343**(1994), no. 2, 479–524. MR**1242787**, DOI 10.1090/S0002-9947-1994-1242787-6 - Tam, B. S., Schneider, H., Linear equations over cones, Collatz-Wielandt numbers and alternating sequences, in preparation.
- Bit Shun Tam and Shiow Fang Wu,
*On the Collatz-Wielandt sets associated with a cone-preserving map*, Linear Algebra Appl.**125**(1989), 77–95. MR**1024484**, DOI 10.1016/0024-3795(89)90033-5 - H. D. Victory Jr.,
*On linear integral operators with nonnegative kernels*, J. Math. Anal. Appl.**89**(1982), no. 2, 420–441. MR**677739**, DOI 10.1016/0022-247X(82)90111-1 - H. D. Victory Jr.,
*The structure of the algebraic eigenspaces to the spectral radius of eventually compact, nonnegative integral operators*, J. Math. Anal. Appl.**90**(1982), no. 2, 484–516. MR**680174**, DOI 10.1016/0022-247X(82)90076-2 - Martin Zerner,
*Quelques propriétés spectrales des opérateurs positifs*, J. Funct. Anal.**72**(1987), no. 2, 381–417 (French). MR**886819**, DOI 10.1016/0022-1236(87)90094-2

## Bibliographic 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
- 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.
- © Copyright 2000 American Mathematical Society
- 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
- MathSciNet review: 1707205