Analytic subordination theory of operator-valued free additive convolution and the solution of a general random matrix problem

References

[1] N. I. Akhieser, The classical moment problem and some related questions in analysis, Hafner Publishing, New York 1965. Suche in Google Scholar

[2] G. W. Anderson, Support properties of spectra of polynomials in Wigner matrices, lecture notes (2012) Suche in Google Scholar

[3] G. W. Anderson, Convergence of the largest singular value of a polynomial in independent Wigner matrices, Ann. Probab. 41 (2013), no. 3B, 2103–2181. 10.1214/11-AOP739Suche in Google Scholar

[4] Z. Bai and J. Silverstein, Spectral analysis of large dimensional random matrices, Math. Monogr. Ser. 2, Science Press, Beijing 2006. Suche in Google Scholar

[5] S. T. Belinschi and H. Bercovici, A new approach to subordination results in free probability, J. Anal. Math. 101 (2007), 357–365. 10.1007/s11854-007-0013-1Suche in Google Scholar

[6] S. T. Belinschi, M. Popa and V. Vinnikov, Infinite divisibility and a non-commutative Boolean-to-free Bercovici–Pata bijection, J. Funct. Anal. 262 (2012), no. 1, 94–123. 10.1016/j.jfa.2011.09.006Suche in Google Scholar

[7] S. T. Belinschi, P. Sniady and R. Speicher, in preparation. Suche in Google Scholar

[8] S. T. Belinschi, R. Speicher, J. Treilhard and C. Vargas, Operator-valued free multiplicative convolution: Analytic subordination theory and applications to random matrix theory, Int. Math. Res. Not. IMRN 2014 (2014), Article ID rnu114. 10.1093/imrn/rnu114Suche in Google Scholar

[9] H. Bercovici and D. V. Voiculescu, Free convolutions of measures with unbounded support, Indiana Univ. Math. J. 42 (1993), no. 3, 733–773. 10.1512/iumj.1993.42.42033Suche in Google Scholar

[10] H. Bercovici and D. V. Voiculescu, Regularity questions for free convolutions, Nonselfadjoint operator algebras, operator theory, and related topics, Oper. Theory Adv. Appl. 104, Birkhäuser, Basel (1998), 37–47. 10.1007/978-3-0348-8779-3_3Suche in Google Scholar

[11] P. Biane, Processes with free increments, Math. Z. 227 (1998), 143–174. 10.1007/PL00004363Suche in Google Scholar

[12] G. Chistyakov and F. Götze, The arithmetic of distributions in free probability theory, Cent. Eur. J. Math. 9 (2011), 997–1050.10.2478/s11533-011-0049-4Suche in Google Scholar

[13] R. Couillet and M. Debbah, Random matrix methods for wireless communications, Cambridge University Press, Cambridge 2011. 10.1017/CBO9780511994746Suche in Google Scholar

[14] S. Dineen, Complex analysis in locally convex spaces, North Holland Math. Stud. 57, North Holland, Amsterdam 1981. Suche in Google Scholar

[15] S. Dineen, The Schwarz lemma, Oxford University Press, Oxford 1989. Suche in Google Scholar

[16] K. Dykema, On the S-transform over a Banach algebra, J. Funct. Anal. 231 (2006), no. 1, 90–110. 10.1016/j.jfa.2005.04.008Suche in Google Scholar

[17] V. L. Girko, Theory of stochastic canonical equations, Math. Appl. (Dordrecht) 535, Kluwer Academic Publishers, Dordrecht 2001. 10.1007/978-94-010-0989-8Suche in Google Scholar

[18] U. Haagerup, H. Schultz and S. Thorbjørnsen, A random matrix approach to the lack of projections in Cred∗⁢(𝔽2), Adv. Math. 204 (2006), 1–83. 10.1016/j.aim.2005.05.008Suche in Google Scholar

[19] U. Haagerup and S. Thorbjørnsen, A new application of random matrices: Ext⁡(Cred∗⁢(F2)) is not a group, Ann. of Math. (2) 162 (2005), no. 2, 711–775. 10.4007/annals.2005.162.711Suche in Google Scholar

[20] L. Harris, Fixed points of holomorphic mappings for domains in Banach spaces, Abstr. Appl. Anal. 5 (2003), 261–274. 10.1155/S1085337503205042Suche in Google Scholar

[21] W. Hechem, P. Loubaton and J. Najim, Deterministic equivalents for certain functionals of large random matrices, Ann. Appl. Probab. 17 (2007), 875–930. 10.1214/105051606000000925Suche in Google Scholar

[22] J. W. Helton, S. A. McCullough and V. Vinnikov, Noncommutative convexity arises from linear matrix inequalities, J. Funct. Anal. 240 (2006), no. 1, 105–191. 10.1016/j.jfa.2006.03.018Suche in Google Scholar

[23] J. W. Helton, R. Rashidi Far and R. Speicher, Operator-valued semicircular elements: Solving a quadratic matrix equation with positivity constraints, Int. Math. Res. Not. IMRN 2007 (2007), no. 22, Article ID rnm086. 10.1093/imrn/rnm086Suche in Google Scholar

[24] F. Hiai and D. Petz, The semicircle law, free random variables and entropy, American Mathematical Society, Providence 2000. Suche in Google Scholar

[25] E. Hille and R. S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ. 31, American Mathematical Society, Providence 1974. Suche in Google Scholar

[26] V. Marchenko and L. Pastur, Distributions of eigenvalues for some sets of random matrices, Math. USSR Sb. 1 (1967), 457–483. 10.1070/SM1967v001n04ABEH001994Suche in Google Scholar

[27] A. Moustakas and S. Simon, Random matrix theory of multi-antenna communications: The Rician channel, J. Phys. A 38 (2005), 10859–10872. 10.1088/0305-4470/38/49/024Suche in Google Scholar

[28] R. Müller, A random matrix model of communications via antenna arrays, IEEE Trans. Inform. Theory 48 (2002), 2495–2506. 10.1109/TIT.2002.801467Suche in Google Scholar

[29] A. Nica and R. Speicher, Commutators of free random variables, Duke Math. J. 92 (1998), no. 3, 553–592. 10.1215/S0012-7094-98-09216-XSuche in Google Scholar

[30] A. Nica and R. Speicher, Lectures on the combinatorics of free probability, Cambridge University Press, Cambridge 2006. 10.1017/CBO9780511735127Suche in Google Scholar

[31] N. Raj Rao and A. Edelman, The polynomial method for random matrices, Found. Comput. Math. 8 (2008), 649–702. 10.1007/s10208-007-9013-xSuche in Google Scholar

[32] M. P. Schützenberger, On the definition of a family of automata, Inform. Control 4 (1961), 245–270. 10.1016/S0019-9958(61)80020-XSuche in Google Scholar

[33] D. Shlyakhtenko, Random Gaussian band matrices and freeness with amalgamation, Int. Math. Res. Not. IMRN 1996 (1996), no. 20, 1013–1025. 10.1155/S1073792896000633Suche in Google Scholar

[34] D. Shlyakhtenko and P. Skoufranis, Freely independent random variables with non-atomic distributions, Trans. Amer. Math. Soc. (2015), 10.1090/S0002-9947-2015-06434-4. 10.1090/S0002-9947-2015-06434-4Suche in Google Scholar

[35] R. Speicher, Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Mem. Amer. Math. Soc. 132 (1998), Paper No. 627. 10.1090/memo/0627Suche in Google Scholar

[36] R. Speicher and C. Vargas, Free deterministic equivalents, rectangular random matrix models, and operator-valued free probability theory, Random Matrices Theory Appl. 1 (2012), Article ID 1150008. 10.1142/S2010326311500080Suche in Google Scholar

[37] A. Tulino and S. Verdù, Random matrix theory and wireless communications, Found. Trends Commun. Inf. Theory 1 (2004), 1–182. 10.1561/0100000001Suche in Google Scholar

[38] V. Vasilchuk, On the asymptotic distribution of the commutator and anticommutator of random matrices, J. Math. Phys. 44 (2003), no. 4, 1882–1908. 10.1063/1.1557329Suche in Google Scholar

[39] D. V. Voiculescu, Symmetries of some reduced free product C∗-algebras, Operator algebras and their connections with topology and ergodic theory (Buşteni 1983), Lecture Notes in Math. 1132, Springer, Berlin (1985), 556–588. 10.1007/BFb0074909Suche in Google Scholar

[40] D. V. Voiculescu, Addition of certain non-commutative random variables, J. Funct. Anal. 66 (1986), 323–346. 10.1016/0022-1236(86)90062-5Suche in Google Scholar

[41] D. V. Voiculescu, Multiplication of certain noncommuting random variables, J. Operator Theory 18 (1987), 223–235. Suche in Google Scholar

[42] D. V. Voiculescu, Limit laws for random matrices and free products, Invent. Math. 104 (1991), 201–220. 10.1007/BF01245072Suche in Google Scholar

[43] D. V. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory. I, Comm. Math. Phys. 155 (1993), 411–440. 10.1007/BF02100050Suche in Google Scholar

[44] D. V. Voiculescu, Operations on certain non-commutative operator-valued random variables, Recent advances in operator algebras (Orléans 1992), Astérisque 232, Société Mathématique de France, Paris (1995), 243–275. Suche in Google Scholar

[45] D. V. Voiculescu, The coalgebra of the free difference quotient and free probability, Int. Math. Res. Not. IMRN 2000 (2000), no. 2, 79–106. 10.1155/S1073792800000064Suche in Google Scholar

[46] D. V. Voiculescu, Analytic subordination consequences of free Markovianity, Indiana Univ. Math. J. 51 (2002), 1161–1166. 10.1512/iumj.2002.51.2252Suche in Google Scholar

[47] D. V. Voiculescu, K. Dykema and A. Nica, Free random variables, CRM Monogr. Ser. 1, American Mathematical Society, Providence 1992. 10.1090/crmm/001Suche in Google Scholar

[48] M. A. Zorn, Characterization of analytic functions in Banach spaces, Ann. of Math. (2) 46 (1945), 585–593. 10.2307/1969198Suche in Google Scholar

[49] M. A. Zorn, Derivatives and Fréchet differentials, Bull. Amer. Math. Soc. 52 (1946), 133–137. 10.1090/S0002-9904-1946-08524-9Suche in Google Scholar