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On the core of a cone-preserving map


Authors: Bit Shun Tam and Hans Schneider
Journal: Trans. Amer. Math. Soc. 343 (1994), 479-524
MSC: Primary 15A48; Secondary 47B65
DOI: https://doi.org/10.1090/S0002-9947-1994-1242787-6
MathSciNet review: 1242787
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Abstract: This is the third of a sequence of papers in an attempt to study the Perron-Frobenius theory of a nonnegative matrix and its generalizations from the cone-theoretic viewpoint. Our main object of interest here is the core of a cone-preserving map. If A is an $ n \times n$ real matrix which leaves invariant a proper cone K in $ {\mathbb{R}^n}$, then by the core of A relative to K, denoted by $ {\text{core}}_K(A)$, we mean the convex cone $ \bigcap\nolimits_{i = 1}^\infty {{A^i}K} $. It is shown that when $ {\text{core}}_K(A)$ is polyhedral, which is the case whenever K is, then $ {\text{core}}_K(A)$ is generated by the distinguished eigenvectors of positive powers of A. The important concept of a distinguished A-invariant face is introduced, which corresponds to the concept of a distinguished class in the nonnegative matrix case. We prove a significant theorem which describes a one-to-one correspondence between the distinguished A-invariant faces of K and the cycles of the permutation induced by A on the extreme rays of $ {\text{core}}_K(A)$, provided that the latter cone is nonzero, simplicial. By an interplay between cone-theoretic and graph-theoretic ideas, the extreme rays of the core of a nonnegative matrix are fully described. Characterizations of K-irreducibility or A-primitivity of A are also found in terms of $ {\text{core}}_K(A)$. Several equivalent conditions are also given on a matrix with an invariant proper cone so that its spectral radius is an eigenvalue of index one. An equivalent condition in terms of the peripheral spectrum is also found on a real matrix A with the Perron-Schaefer condition for which there exists a proper invariant cone K suchthat $ {\text{core}}_K(A)$ is polyhedral, simplicial, or a single ray. A method of producing a large class of invariant proper cones for a matrix with the Perron-Schaefer condition is also offered.


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  • [Bar 1] G. P. Barker, On matrices having an invariant cone, Czechoslovak Math. J. 22 (1972), 49-68. MR 0302672 (46:1816)
  • [Bar 2] -, Theory of cones, Linear Algebra Appl. 39 (1981), 263-291. MR 625256 (83e:15022)
  • [B-N] R. Bru and M. Neumann, Nonnegative Jordan bases, Linear and Multilinear Algebra 23 (1988), 95-109. MR 966618 (90a:15024)
  • [B-N-S] A. Berman, M. Neumann, and R. J. Stern, Nonnegative matrices in dynamical system, J. Wiley, New York, 1989. MR 1019319 (90j:93030)
  • [B-P] A. Berman and R. J. Plemmons, Nonnegative matrices in the mathematical sciences, Academic Press, New York, 1979. MR 544666 (82b:15013)
  • [B-S] G. P. Barker and H. Schneider, Algebraic Perron-Frobenius theory, Linear Algebra Appl. 11 (1975), 219-233. MR 0374171 (51:10371)
  • [B-T] G. P. Barker and R. E. L. Turner, Some observations on the spectra of cone-preserving maps, Linear Algebra Appl. 6 (1973), 149-153. MR 0325636 (48:3983)
  • [B-T-D] G. P. Barker, B. S. Tam, and N. Davila, A geometric Gordan-Stiemke theorem, Linear Algebra Appl. 61 (1984), 83-89. MR 755250 (86j:90088)
  • [Bir] G. Birkhoff, Linear transformations with invariant cones, Amer. Math. Monthly 74 (1967), 274-276. MR 0214605 (35:5454)
  • [Bon] F. F. Bonsall, Sublinear functionals and ideals in partially ordered vector spaces, Proc. London Math. Soc. 4 (1954), 402-418. MR 0068752 (16:936c)
  • [Dok] D. Ž. Djoković, Quadratic cones invariant under some linear operators, SIAM J. Algebraic Discrete Methods 8 (1987), 186-191. MR 881178 (88i:15042)
  • [Ell] A. J. Ellis, On faces of compact convex sets and their annihilators, Math. Ann. 184 (1969), 19-24. MR 0253012 (40:6227)
  • [Els 1] L. Eisner, Monotonie und Randspektrum bei vollstetigen Operatoren, Arch. Rational Mech. Anal. 36 (1970), 356-365. MR 0251577 (40:4804)
  • [Els 2] -, On matrices leaving invariant a nontrivial convex set, Linear Algebra Appl. 42 (1982), 103-107. MR 656417 (83k:15016)
  • [Fri 1] S. Friedland, Characterizations of spectral radius of positive operators, Linear Algebra Appl. 134 (1990), 93-105. MR 1060012 (92d:47054)
  • [Fri 2] -, Characterizations of spectral radius of positive operators on $ {C^\ast}$ algebras, J. Funct. Anal. 97 (1991), 64-70. MR 1105655 (92g:46083)
  • [Fro] G. F. Frobenius, Über Matrizen aus nicht negativen Elementen, Sitzungsber. Kon. Preuss Akad. Wiss. Berlin, (1912), 456-477; Ges. Abh. 3 (1968), 546-567.
  • [G-L-R] I. Gohberg, P. Lancaster, and L. Rodman, Invariant subspaces of matrices with applications, J. Wiley, New York, 1986. MR 873503 (88a:15001)
  • [Gro] U. Groh, Some observations on the spectra of positive operators on finite-dimensional $ {C^\ast}$algebras, Linear Algebra Appl. 42 (1982), 213-222. MR 656426 (84f:46079)
  • [H-S 1] D. Hershkowitz and H. Schneider, Height bases, level bases, and the equality of height and level characteristics of an M-matrix, Linear and Multilinear Algebra 25 (1989), 149-171. MR 1018774 (90k:15006)
  • [H-S 2] -, Combinatorial basis, derived Jordan sets and the equality of the height and the level characteristics of an M-matrix, Linear and Multilinear Algebra 29 (1991), 21-42. MR 1118353 (92h:15008)
  • [H-S 3] -, On the existence of matrices with prescribed height and level characteristics, Israel J. Math. 75 (1991), 105-117. MR 1147293 (92m:15017)
  • [H-W] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 4th ed., Oxford Univ. Press, London, 1960.
  • [Hor] J. G. Horne, On the automorphism group of a cone, Linear Algebra Appl. 21 (1978), 111-121. MR 0491812 (58:11009b)
  • [J-V] R. J. Jang and H. D. Victory, Jr., On nonnegative solvability of linear integral equations, preprint.
  • [K-R] M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Amer. Math. Soc. Transl. Ser. (1) 10 (1962), 199-325 [originally Uspekhi Mat. Nauk (N.S.) 3 (1948), 3-95]. MR 0027128 (10:256c)
  • [L-T] P. Lancaster and M. Tismenetsky, The theory of matrices, 2nd ed., Academic Press, New York, 1985. MR 792300 (87a:15001)
  • [Min] H. Minc, Nonnegative matrices, Wiley, New York, 1988. MR 932967 (89i:15001)
  • [Nel] P. Nelson, Jr., The structure of a positive linear integral operator, J. London Math. Soc. (2) 8 (1974), 711-718. MR 0355695 (50:8169)
  • [Pul] N. J. Pullman, A geometric approach to the theory of nonnegative matrices, Linear Algebra Appl. 4 (1971), 297-312. MR 0286816 (44:4023)
  • [Roc] R. T. Rockafellar, Convex analysis, Princeton Univ. Press, Princeton, NJ, 1970. MR 0274683 (43:445)
  • [Rot] U. G. Rothblum, Algebraic eigenspaces of nonnegative matrices, Linear Algebra Appl. 12 (1975), 281-292. MR 0404298 (53:8100)
  • [Scha 1] H. H. Schaefer, Banach lattices and positive operators, Springer, New York, 1974. MR 0423039 (54:11023)
  • [Scha 2] -, Topological vector spaces, 4th printing, Springer, New York, 1980.
  • [Scha 3] -, A metric variant of Frobenius theorem and some other remarks on positive matrices, Linear Algebra Appl. 42 (1982), 175-182. MR 656423 (83h:15023)
  • [Schn 1] H. Schneider, Geometric conditions for the existence of positive eigenvalues of matrices, Linear Algebra Appl. 38 (1981), 253-271. MR 636041 (83f:15012)
  • [Schn 2] -, 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 (1986), 161-189. MR 872282 (88b:15010)
  • [S-W 1] R. Stern and H. Wolkowicz, Invariant ellipsoidal cones, Linear Algebra Appl. 150 (1991), 81-106. MR 1102058 (92h:15014)
  • [S-W 2] -, A note on generalized invariant cones and the Kronecker canonical form, Linear Algebra Appl. 147 (1991), 97-100. MR 1088661 (92e:15018)
  • [Tam 1] B. S. Tam, On the distinguished eigenvalues of a cone-preserving map, Linear Algebra Appl. 131 (1990), 17-37. MR 1057062 (91d:15040)
  • [Tam 2] -, On matrices with invariant proper cones, (in preparation).
  • [T-S] B. S. Tam and H. Schneider, On the invariant faces of a cone-preserving map, (in preparation).
  • [T-W] B. S. Tam and S. F. Wu, On the Collatz-Wielandt sets associated with a cone-preserving map, Linear Algebra Appl. 125 (1989), 77-95. MR 1024484 (90i:15021)
  • [Van] J. S. Vandergraft, Spectral properties of matrices which have invariant cones, SIAM J. Appl. Math. 16 (1968), 1208-1222. MR 0244284 (39:5599)
  • [V-L] A. I. Veitsblit and Yu. I. Lyubich, Boundary spectrum of nonnegative operators, Siberian Math. J. 26 (1985), 798-802 [translated from Sibirsk. Mat. Zh. 26 (1985), 24-28]. MR 816500 (87e:47046)
  • [Vic] H. D. Victory, Jr., On linear integral operators with nonnegative kernels, J. Math. Anal. Appl. 29 (1982), 420-441. MR 677739 (84j:45003)
  • [Vic 2] -, The structure of the algebraic eigenspace to the spectral radius of eventually compact, nonnegative integral operators, J. Math. Anal. Appl. 90 (1982), 484-516. MR 680174 (84d:47041)
  • [Vic 3] -, On nonnegative solutions to matrix equations, SIAM J. Algebraic Discrete Methods 6 (1985), 406-412. MR 791170 (86h:15003)

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DOI: https://doi.org/10.1090/S0002-9947-1994-1242787-6
Keywords: Cone-preserving map, core, distinguished eigenvalue, distinguished invariant face, nonnegative matrix, peripheral spectrum, Perron-Schaefer condition, polyhedral cone, simplicial cone, spectral radius of index one
Article copyright: © Copyright 1994 American Mathematical Society

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