Skip to main content


  • TAREK HAMDI (a1) (a2)

In this paper, we perform a detailed spectral study of the liberation process associated with two symmetries of arbitrary ranks: $(R,S)\mapsto (R,U_{t}SU_{t}^{\ast })_{t\geqslant 0}$ , where $(U_{t})_{t\geqslant 0}$ is a free unitary Brownian motion freely independent from $\{R,S\}$ . Our main tool is free stochastic calculus which allows to derive a partial differential equation (PDE) for the Herglotz transform of the unitary process defined by $Y_{t}:=RU_{t}SU_{t}^{\ast }$ . It turns out that this is exactly the PDE governing the flow of an analytic function transform of the spectral measure of the operator $X_{t}:=PU_{t}QU_{t}^{\ast }P$ where $P,Q$ are the orthogonal projections associated to $R,S$ . Next, we relate the two spectral measures of $RU_{t}SU_{t}^{\ast }$ and of $PU_{t}QU_{t}^{\ast }P$ via their moment sequences and use this relationship to develop a theory of subordination for the boundary values of the Herglotz transform. In particular, we explicitly compute the subordinate function and extend its inverse continuously to the unit circle. As an application, we prove the identity $i^{\ast }(\mathbb{C}P+\mathbb{C}(I-P);\mathbb{C}Q+\mathbb{C}(I-Q))=-\unicode[STIX]{x1D712}_{\text{orb}}(P,Q)$ .

Hide All
[1] Belinschi, S. T. and Bercovici, H., Partially defined semigroups relative to multiplicative free convolution , Int. Math. Res. Not. IMRN 2 (2005), 65101.
[2] Biane, P., Free Brownian Motion, Free Stochastic Calculus and Random Matrices, Fields Inst. Commun. 12 , Amer. Math. Soc., Providence, RI, 1997, 119.
[3] Cima, J., Matheson, A. L. and Ross, W. T., The Cauchy Transform, Mathematical Surveys and Monographs 125 , Amer. Math. Soc., Providence, RI, 2006.
[4] Collins, B. and Kemp, T., Liberation of projections , J. Funct. Anal. 266 (2014), 19882052.
[5] Conway, J. B., Functions of One Complex Variable II, Graduate Texts in Mathematics, Springer-Verlag, New York, 1995.
[6] Demni, N. and Hamdi, T., Inverse of the flow and moments of the free Jacobi process associated with one projection , Random Matrices: Theory Appl. (2) 07 (2018), 1850001.
[7] Demni, N., Hamdi, T. and Hmidi, T., The spectral distribution of the free Jacobi process , Indiana Univ. Math. J. 61(3) (2012), 13511368.
[8] Demni, N. and Hmidi, T., Spectral distribution of the free Jacobi process associated with one projection , Colloq. Math. 137(2) (2014), 271296.
[9] Duren, P. L., Theory of H p Spaces, Dover, Mineola, New York, 2000.
[10] Dykema, K. J., Nica, A. and Voiculescu, D. V., Free Random Variables, CRM Monograph Series 1 , American Mathematical Society, 1992.
[11] Garnett, J. B., Bounded Analytic Functions, Pure and Applied Mathematics 96 , Academic Press, New York, 1981.
[12] Hiai, F., Miyamoto, T. and Ueda, Y., Orbital approach to microstate free entropy , Internat. J. Math. 20 (2009), 227273.
[13] Hiai, F. and Petz, D., Large deviations for functions of two random projection matrices , Acta Sci. Math. (Szeged) 72 (2006), 581609.
[14] Hiai, F. and Ueda, Y., A log-Sobolev type inequality for free entropy of two projections , Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009), 239249.
[15] Izumi, M. and Ueda, Y., Remarks on free mutual information and orbital free entropy , Nagoya Math. J 220 (2015), 4566.
[16] Lawler, G. F., Conformally Invariant Processes in the Plane, Mathematical Surveys and Monographs 114 , Americal Mathematical Society, Providence, RI, 2005.
[17] Nica, A. and Speicher, R., Lectures on the Combinatorics of Free Probability, Lecture Note Series 335 , London Mathematical Society, Cambridge University Press, 2006.
[18] Ueda, Y., Orbital free entropy, revisited , Indiana Univ. Math. J. 63 (2014), 551577.
[19] Voiculescu, D. V., The analogues of entropy and of Fisher’s information measure in free probability theory. VI. Liberation and mutual free information , Adv. Math. 146(2) (1999), 101166.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Nagoya Mathematical Journal
  • ISSN: 0027-7630
  • EISSN: 2152-6842
  • URL: /core/journals/nagoya-mathematical-journal
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed