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Research Article

Embeddability of real and positive operators

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Pages 3747-3767 | Received 14 Apr 2020, Accepted 15 Nov 2020, Published online: 08 Dec 2020

References

  • Elfving G. Zur Theorie der Markoffschen Ketten. Acta Soc Sci Fennicae A 2. 1937;8:1–17.
  • Kingman J. The imbedding problem for finite Markov chains. Z Wahrsch Verw Gebiete. 1962;1:14–24.
  • Davies EB. Embeddable Markov matrices. Electron J Probab. 2010;15:1474–1486.
  • Baake M, Sumner J. Notes on Markov embedding. Linear Algebra Appl. 2020;594:262–299.
  • Veerman JJP, Kummel E. Diffusion and consensus on weakly connected directed graphs. Linear Algebra Appl. 2019;578:184–206.
  • Singer B, Spilerman S. The representation of social processes by Markov models. Am J Sociol. 1976;82:1–54.
  • Verbyla KL, Yap VB, Pahwa A, et al. The embedding problem for Markov models of nucleotide substitution. PLoS One. 2013;8(7):e69187.
  • Israel RB, Rosenthal JS, Wei JZ. Finding generators for Markov chains via empirical transition matrices, with applications to credit ratings. Math Finance. 2001;11:245–265.
  • King JL. The generic transformation has roots of all orders. Colloq Math. 2000;84/85:521–547.
  • de la Rue T. A generic transformation can be embedded in a flow. Ann Inst H Poincaré Prob Stat. 2003;39:121–134.
  • Stepin AM, Eremenko AM. Nonuniqueness of an inclusion in a flow and the vastness of a centralizer for a generic measure-preserving transformation. Mat Sb. 2004;195:95–108; translation in Sb. Math. 2004;195:1795–1808.
  • Heyer H. Probability measures on locally compact groups. Berlin: Springer-Verlag; 1977.
  • Fisher MJ. The embeddability of an invertible measure. Semigroup Forum. 1972/73;5:340–353.
  • Haase M. The functional calculus for sectorial operators. Basel: Birkhäuser Verlag; 2006. (Operator theory: advances and applications; vol. 169).
  • Haase M. Functional calculus for groups and applications to evolution equations. J Evol Equ. 2007;7:529–554.
  • Eisner T. Embedding operators into strongly continuous semigroups. Arch Math (Basel). 2009;92:451–460.
  • Cubitt TS, Eisert J, Wolf MM. The complexity of relating quantum channels to master equations. Commun Math Phys. 2012;310:383–418.
  • Müller-Hermes A, Reeb D, Wolf MM. Quantum subdivision capacities and continuous-time quantum coding. IEEE Trans Inform Theory. 2015;61:565–581.
  • Engel K-J, Nagel R. One-parameter semigroups for linear evolution equations. New York: Springer; 2000. (Graduate texts in mathematics; 194).
  • Speakman JMO. Two Markov chains with a common skeleton. Z Wahrsch Verw Gebiete. 1967;7:224.
  • Cuthbert JR. The logarithm function of finite-state Markov semi-groups. J Lond Math Soc. 1973;6:524–532.
  • Johansen S. Some results on the imbedding problem for finite Markov chains. J Lond Math Soc. 1974;8:345–351.
  • Carette P. Characterizations of embeddable 3 × 3 stochastic matrices with a negative eigenvalue. New York J. Math.. 1994/95;1:120–129, electronic.
  • Casanellas M, Fernández-Sánchez J, Roca-Lacostena J. The embedding problem for Markov matrices; 2020. Preprint arXiv:2005.00818.
  • Culver WJ. On the existence and uniqueness of the real logarithm of a matrix. Proc Am Math Soc. 1966;17:1146–1151.
  • Horn R, Johnson C. Matrix analysis. Cambridge: Cambridge University Press; 1990.
  • Singer B, Spilerman S. Social mobility models for heterogeneous populations. Sociol. Method. 1973–1974;5:356–401.
  • Guerry MA. On the embedding problem for discrete-time Markov chains. J Appl Probab. 2013;50:918–930.
  • Guerry MA. Some results on the embeddable problem for discrete-time Markov models in manpower planning. Commun Stat Theory Methods. 2014;43:1575–1584.
  • Higham NJ, Lin L. On pth roots of stochastic matrices. Linear Algebra Appl. 2011;435:448–463.
  • Johnson CR. Inverse M-matrices. Linear Algebra Appl. 1982;47:195–216.
  • Van-Brunt A. Infinitely divisible nonnegative matrices, M-matrices, and the embedding problem for finite state stationary Markov chains. Linear Algebra Appl. 2018;541:163–176.
  • Bausch J, Cubitt T. The complexity of divisibility. Linear Algebra Appl. 2016;504:64–107.
  • Higham NJ. Functions of matrices: theory and computation. Philadelphia (PA): Society for Industrial and Applied Mathematics (SIAM); 2008.
  • Higham NJ. Computing real square roots of a real matrix. Linear Algebra Appl. 1987;88/89:405–430.
  • Muñoz GA, Sarantopoulos Y, Tonge A. Complexifications of real Banach spaces, polynomials and multilinear maps. Studia Math. 1999;134:1–33.
  • Schaefer HH. Banach lattices and positive operators. Springer; 1974.
  • Keicher V. On the peripheral spectrum of bounded positive semigroups on atomic Banach lattices. Arch Math (Basel). 2006;87:359–367.
  • Davies EB. Triviality of the peripheral point spectrum. J Evol Equ. 2005;5:407–415.
  • Wolff MPH. Triviality of the peripheral point spectrum of positive semigroups on atomic Banach lattices. Positivity. 2008;12:185–192.
  • Glück J. Spectral and asymptotic properties of contractive semigroups on non-Hilbert spaces. J Oper Theory. 2016;76:3–31.
  • Nagel R (ed.). One-parameter semigroups of positive operators. Berlin: Springer-Verlag; 1986. (Lecture notes in mathematics; 1184).
  • Bátkai A, Kramar Fijavž M, Rhandi A. Positive operator semigroups: from finite to infinite dimensions. Basel: Birkhäuser; Cham: Springer; 2017. (Operator theory: advances and applications; 257).
  • Tam B-S, Huang P-R. Nonnegative square roots of matrices. Linear Algebra Appl. 2016;498:404–440.
  • Kechris AS. Classical descriptive set theory. New York: Springer-Verlag; 1995. (Graduate texts in mathematics; 156).
  • Eisner T, Mátrai T. On typical properties of Hilbert space operators. Israel J Math. 2013;195:247–281.

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