The Collatz-Wielandt quotient for pairs of nonnegative operators. (English) Zbl 07285946

Summary: In this paper we consider two versions of the Collatz-Wielandt quotient for a pair of nonnegative operators \(A,B\) that map a given pointed generating cone in the first space into a given pointed generating cone in the second space. If the two spaces and two cones are identical, and \(B\) is the identity operator, then one version of this quotient is the spectral radius of \(A\). In some applications, as commodity pricing, power control in wireless networks and quantum information theory, one needs to deal with the Collatz-Wielandt quotient for two nonnegative operators. In this paper we treat the two important cases: a pair of rectangular nonnegative matrices and a pair of completely positive operators. We give a characterization of minimal optimal solutions and polynomially computable bounds on the Collatz-Wielandt quotient.


15A22 Matrix pencils
15A45 Miscellaneous inequalities involving matrices
15B48 Positive matrices and their generalizations; cones of matrices
15B57 Hermitian, skew-Hermitian, and related matrices
94A40 Channel models (including quantum) in information and communication theory
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[1] Avin, C.; Borokhovich, M.; Haddad, Y.; Kantor, E.; Lotker, Z.; Parter, M.; Peleg, D., Generalized Perron-Frobenius theorem for multiple choice matrices, and applications, Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013 SIAM, Philadelphia (2013), 478-497 · Zbl 1422.90013
[2] Avin, C.; Borokhovich, M.; Haddad, Y.; Kantor, E.; Lotker, Z.; Parter, M.; Peleg, D., Generalized Perron-Frobenius theorem for nonsquare matrices, Available at https://arxiv.org/abs/1308.5915 (2013), 55 pages · Zbl 1360.65107
[3] Berman, A.; Plemmons, R. J., Nonnegative Matrices in Mathematical Sciences, Computer Science and Applied Mathematics. Academic Press, New York (1979) · Zbl 0484.15016
[4] Boutry, G.; Elad, M.; Golub, G. H.; Milanfar, P., The generalized eigenvalue problem for nonsquare pencils using a minimal perturbation approach, SIAM J. Matrix Anal. Appl. 27 (2005), 582-601 · Zbl 1100.65035
[5] Boyd, S.; Vandenberghe, L., Convex Optimization, Cambridge University Press, New York (2004) · Zbl 1058.90049
[6] Choi, M.-D., Completely positive linear maps on complex matrices, Linear Algebra Appl. 10 (1975), 285-290 · Zbl 0327.15018
[7] Chu, D.; Golub, G. H., On a generalized eigenvalue problem for nonsquare pencils, SIAM J. Matrix Anal. Appl. 28 (2006), 770-787 · Zbl 1128.15004
[8] Collatz, L., Einschliessungssatz für die charakteristischen Zahlen von Matrizen, Math. Z. 48 (1942), 221-226 German · Zbl 0027.00604
[9] Erdelyi, I., On the matrix equation \(Ax=\lambda Bx\), J. Math. Anal. Appl. 17 (1967), 119-132 · Zbl 0153.04902
[10] Friedland, S., Characterizations of the spectral radius of positive operators, Linear Algebra Appl. 134 (1990), 93-105 · Zbl 0707.15005
[11] Friedland, S., Characterizations of spectral radius of positive operators on \(C^*\) algebras, J. Funct. Anal. 97 (1991), 64-70 · Zbl 0745.47024
[12] Friedland, S., Matrices: Algebra, Analysis and Applications, World Scientific, Hackensack (2016) · Zbl 1337.15002
[13] Friedland, S.; Loewy, R., On the extreme points of quantum channels, Linear Algebra Appl. 498 (2016), 553-573 · Zbl 1334.15086
[14] Frobenius, G. F., Über Matrizen aus positiven Elementen, Berl. Ber. 1908 (1908), 471-476 German \99999JFM99999 39.0213.03 · JFM 39.0213.03
[15] Frobenius, G. F., Über Matrizen aus positiven Elementen II, Berl. Ber. 1909 (1909), 514-518 German \99999JFM99999 40.0202.02 · JFM 40.0202.02
[16] Frobenius, G. F., Über Matrizen aus nicht negativen Elementen, Berl. Ber. 1912 (1912), 456-477 German \99999JFM99999 43.0204.09 · JFM 43.0204.09
[17] Gantmacher, F. R., The Theory of Matrices. Vol. 1, Chelsea Publishing, New York (1959) · Zbl 0927.15001
[18] Gantmacher, F. R., The Theory of Matrices. Vol. 2, Chelsea Publishing, New York (1959) · Zbl 0927.15002
[19] Golub, G. H.; Loan, C. F. Van, Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore (2013) · Zbl 1268.65037
[20] Grötschel, M.; Lovász, L.; Schrijver, A., Geometric Algorithms and Combinatorial Optimization, Algorithms and Combinatorics 2. Springer, Berlin (1988) · Zbl 0634.05001
[21] Hastings, M. B., Superadditivity of communication capacity using entangled inputs, Nature Phys. 5 (2009), 255-257
[22] Holevo, A. S., The additivity problem in quantum information theory, Proceedings of the International Congress of Mathematicians (ICM). Vol. III European Mathematical Society, Zürich (2006), 999-1018 · Zbl 1100.94007
[23] Holevo, A. S., Quantum Systems, Channels, Information: A Mathematical Introduction, De Gruyter Studies in Mathematical Physics 16. De Gruyter, Berlin (2012) · Zbl 1332.81003
[24] Horn, R. A.; Johnson, C. R., Matrix Analysis, Cambridge University Press, Cambridge (2013) · Zbl 1267.15001
[25] Karlin, S., Positive operators, J. Math. Mech. 8 (1959), 905-937 · Zbl 0087.11002
[26] Kreĭn, M. G.; Rutman, M. A., Linear operators leaving invariant cone in a Banach space, Usp. Mat. Nauk 3 (1948), 3-95 Russian · Zbl 0030.12902
[27] Lovász, L., An Algorithmic Theory of Numbers, Graphs and Convexity, CBMS-NSF Regional Conference Series in Applied Mathematics 50. SIAM, Philadelphia (1986) · Zbl 0606.68039
[28] Mangasarian, O. L., Perron-Frobenius properties of \(Ax-\lambda Bx\), J. Math. Anal. Appl. 36 (1971), 86-102 · Zbl 0224.15010
[29] Mendl, C. B.; Wolf, M. M., Unital quantum channels - convex structure and revivals of Birkhoff’s theorem, Commun. Math. Phys. 289 (2009), 1057-1086 · Zbl 1167.81011
[30] Meyer, C. D., Matrix Analysis and Applied Linear Algebra, SIAM, Philadelphia (2000) · Zbl 0962.15001
[31] Minc, H., Nonnegative Matrices, Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York (1988) · Zbl 0638.15008
[32] Perron, O., Zur Theorie der Matrices, Math. Ann. 64 (1907), 248-263 German \99999JFM99999 38.0202.01 · JFM 38.0202.01
[33] Petz, D., Quantum Information Theory and Quantum Statistics, Theoretical and Mathematical Physics. Springer, Berlin (2008) · Zbl 1145.81002
[34] Pillai, S. U.; Suel, T.; Cha, S., The Perron-Frobenius theorem: Some of its applications, IEEE Signal Process. Magazine 22 (2005), 62-75
[35] Schaeffer, H. H., Banach Lattices and Positive Operators, Die Grundlehren der mathematischen Wissenschaften 215. Springer, Berlin (1974) · Zbl 0296.47023
[36] Seneta, E., Non-Negative Matrices and Markov Chains, Springer Series in Statistics. Springer, New York (1981) · Zbl 0471.60001
[37] Shirokov, M. E., On the structure of optimal sets for a quantum channel, Probl. Inf. Transm. 42 (2006), 282-297 Translation from Probl. Peredachi Inf. 42 2006 23-40 · Zbl 1237.94039
[38] Shor, P. W., Additivity of the classical capacity of entanglement-breaking quantum channels, J. Math. Phys. 43 (2002), 4334-4340 · Zbl 1060.94004
[39] Srikant, R., The Mathematics of Internet Congestion Control, Systems and Control: Foundations and Applications. Birkhäuser, Boston (2004) · Zbl 1086.68018
[40] Wielandt, H., Unzerlegbare, nicht-negative Matrizen, Math. Z. 52 (1950), 642-648 German · Zbl 0035.29101
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