Abstract
The Brown measure is a generalization of the eigenvalue distribution for a general (not necessarily normal) operator in a finite von Neumann algebra (i.e. a von Neumann algebra which possesses a trace). It was introduced by Larry Brown in [46], but fell into obscurity soon after. It was revived by Haagerup and Larsen [85] and played an important role in Haagerup’s investigations around the invariant subspace problem [87].
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References
L. Aagaard, U. Haagerup, Moment formulas for the quasi-nilpotent DT-operator. Int. J. Math. 15(6), 581–628 (2004)
D.H. Armitage, S.J. Gardiner, Classical Potential Theory (Springer, London, 2001)
S.T. Belinschi, P. Śniady, R. Speicher, Eigenvalues of non-hermitian random matrices and Brown measure of non-normal operators: Hermitian reduction and linearization method. arXiv:1506.02017 (2015)
P. Biane, F. Lehner, Computation of some examples of Brown’s spectral measure in free probability. Colloq. Math. 90(2), 181–211 (2001)
C. Bordenave, D. Chafaï, Around the circular law. Probab. Surv. 9, 1–89 (2012)
L.G. Brown, Lidskiĭ’s theorem in the type II case, in Geometric Methods in Operator Algebras (Kyoto, 1983). Pitman Research Notes in Mathematics Series, vol. 123 (Longman Science Technology, Harlow, 1986), pp. 1–35
J. Feinberg, A. Zee, Non-Hermitian random matrix theory: method of Hermitian reduction. Nuclear Phys. B 504(3), 579–608 (1997)
B. Fuglede, R.V. Kadison, Determinant theory in finite factors. Ann. Math. (2) 55, 520–530 (1952)
V.L. Girko, The circular law. Teor. Veroyatnost. i Primenen. 29(4), 669–679 (1984)
A. Guionnet, M. Krishnapur, O. Zeitouni, The single ring theorem. Ann. Math. (2) 174(2), 1189–1217 (2011)
U. Haagerup, F. Larsen, Brown’s spectral distribution measure for R-diagonal elements in finite von Neumann algebras. J. Funct. Anal. 176(2), 331–367 (2000)
U. Haagerup, H. Schultz, Brown measures of unbounded operators affiliated with a finite von Neumann algebra. Math. Scand. 100(2), 209–263 (2007)
U. Haagerup, H. Schultz, Invariant subspaces for operators in a general II1-factor. Publ. Math. Inst. Hautes Études Sci. 109(1), 19–111 (2009)
W.K. Hayman, P.B. Kennedy, Subharmonic Functions. Vol. I. London Mathematical Society Monographs, vol. 9 (Academic, London/New York, 1976)
J.W. Helton, T. Mai, R. Speicher, Applications of realizations (aka linearizations) to free probability. arXiv preprint arXiv:1511.05330, 2015
R.A. Janik, M.A. Nowak, G. Papp, I. Zahed, Non-Hermitian random matrix models. Nuclear Phys. B 501(3), 603–642 (1997)
F. Larsen, Brown measures and R-diagonal elements in finite von Neumann algebras. Ph.D. thesis, University of Southern Denmark, 1999
A. Nica, R. Speicher, R-diagonal pairs—a common approach to Haar unitaries and circular elements, in Free Probability Theory (Waterloo, ON, 1995). Fields Institute Communications, vol. 12 (American Mathematical Society, Providence, RI, 1997), pp. 149–188
A. Nica, R. Speicher, Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series, vol. 335 (Cambridge University Press, Cambridge, 2006)
P. Śniady, R. Speicher, Continuous family of invariant subspaces for R–diagonal operators. Invent. Math. 146(2), 329–363 (2001)
T. Tao, Topics in Random Matrix Theory, vol. 132 (American Mathematical Society Providence, RI, 2012)
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Mingo, J.A., Speicher, R. (2017). Brown Measure. In: Free Probability and Random Matrices. Fields Institute Monographs, vol 35. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-6942-5_11
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