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Resolvents of $ \mathscr{R}$-diagonal operators


Authors: Uffe Haagerup, Todd Kemp and Roland Speicher
Journal: Trans. Amer. Math. Soc. 362 (2010), 6029-6064
MSC (2010): Primary 46L54
DOI: https://doi.org/10.1090/S0002-9947-2010-05140-2
Published electronically: July 7, 2010
MathSciNet review: 2661507
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Abstract: We consider the resolvent $ (\lambda-a)^{-1}$ of any $ \mathscr{R}$-diagonal operator $ a$ in a $ \mathrm{II}_1$-factor. Our main theorem (Theorem 1.1) gives a universal asymptotic formula for the norm of such a resolvent. En route to its proof, we calculate the $ \mathscr{R}$-transform of the operator $ \vert\lambda-c\vert^2$ where $ c$ is Voiculescu's circular operator, and we give an asymptotic formula for the negative moments of $ \vert\lambda-a\vert^2$ for any $ \mathscr{R}$-diagonal $ a$. We use a mixture of complex analytic and combinatorial techniques, each giving finer information where the other can give only coarse detail. In particular, we introduce partition structure diagrams in Section 4, a new combinatorial structure arising in free probability.


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  • 1. Biane, P.; Lehner, F.: Computation of some examples of Brown's spectral measure in free probability. Colloq. Math. 90, 181-211 (2001) MR 1876844 (2003f:46103)
  • 2. Chou, E.; Fricano, A.; Kemp, T.; Poh, J.; Shore, W.; Whieldon, G.; Wong, T.; Zhang, Y.: Convex posets in non-crossing pairings on bitstrings. Preprint.
  • 3. Haagerup, U.: An example of a nonnuclear $ C\sp{*} $-algebra, which has the metric approximation property. Invent. Math. 50 279-293 (1978/79) MR 520930 (80j:46094)
  • 4. Haagerup, U.: Random matrices, free probability and the invariant subspace problem relative to a von Neumann algebra. Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), 273-290 MR 1989189 (2004e:46002)
  • 5. Haagerup, U.; Larsen, F.: Brown's spectral distribution measure for $ \mathscr{R}$-diagonal elements in finite von Neumann algebras. J. Funct. Anal. 176, 331-367 (2000) MR 1784419 (2001i:46106)
  • 6. Haagerup, U.; Schultz, H.: Brown measures of unbounded operators affiliated with a finite von Neumann algebra. Math. Scand. 100, 209-263 (2007) MR 2339369 (2008m:46139)
  • 7. Haagerup, U.; Schultz, H.: Invariant subspaces for operators in a general $ \mathrm{II}_1$-factor. Publ. Math. Inst. Hautes Études Sci., no. 109 (2009), 19-111. MR 2511586
  • 8. Hiwatashi, O.; Kuroda, T.; Nagisa, M.; Yoshida, H.: The free analogue of noncentral chi-square distributions and symmetric quadratic forms in free random variables. Math. Z. 230 (1999), no. 1, 63-77. MR 1671858 (99m:46155)
  • 9. Kemp, T.: $ \mathscr{R}$-diagonal dilation semigroups. Math. Z. 264 (2010), no. 1, 111-136. MR 2564935
  • 10. Kemp, T.; Marlburg, K.; Rattan, A.; Smyth, C.: Enumeration of non-crossing pairings on binary strings. To appear in Jour. Comb. Th. A.
  • 11. Kemp, T.; Speicher, R.: Strong Haagerup inequalities for free $ \mathscr{R}$-diagonal elements. J. Funct. Anal. 251 no. 1, 141-173 (2007) MR 2353703 (2008k:46195)
  • 12. Nica, A.; Shlyakhtenko, D.; Speicher, R.: Maximality of the microstates free entropy for $ \mathscr{R}$-diagonal elements. Pacific J. Math. 187 no. 2, 333-347 (1999) MR 1675037 (2000b:46121)
  • 13. Larsen, F.: Powers of $ R$-diagonal elements. J. Operator Theory 47, no. 1, 197-212 (2002) MR 1905821 (2003g:46077)
  • 14. Nica, A.; Speicher, R.: $ \mathscr{R}$-diagonal pairs--a common approach to Haar unitaries and circular elements. Fields Inst. Commun. 12, 149-188 (1997) MR 1426839 (98b:46083)
  • 15. Nica, A.; Speicher, R.: Commutators of free random variables. Duke Math. J. 92, 553-592 (1998) MR 1620518 (99d:46084)
  • 16. Nica, A.; Speicher, R.: Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series, no. 335, Cambridge University Press, 2006 MR 2266879 (2008k:46198)
  • 17. Przytycki, J.; Sikora, A.: Polygon dissections and Euler, Fuss, Kirkman, and Cayley numbers. J. Combin. Theory Ser. A 92, 68-76 (2000) MR 1783940 (2001g:05005)
  • 18. Śniady, P.; Speicher, R.: Continuous family of invariant subspaces for $ \mathscr{R}$-diagonal operators. Invent. Math. 146, no. 2, 329-363 (2001) MR 1865398 (2002i:47009)
  • 19. Voiculescu, D.: The $ K$-group of the $ C^\ast$-algebra of a semicircular family. K-Theory 7, 5-7 (1997) MR 1220422 (94f:46093)

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Additional Information

Uffe Haagerup
Affiliation: Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55 DK-5230 Odense M, Denmark
Email: haagerup@imada.sdu.dk

Todd Kemp
Affiliation: Department of Mathematics, University of California, San Diego, La Jolla, California 92093
Email: tkemp@math.ucsd.edu

Roland Speicher
Affiliation: Department of Mathematics and Statistics, Queen’s University, Jeffery Hall, Kingston, Ontario, Canada K7L3N6
Email: speicher@mast.queensu.ca

DOI: https://doi.org/10.1090/S0002-9947-2010-05140-2
Received by editor(s): November 24, 2008
Received by editor(s) in revised form: March 3, 2009
Published electronically: July 7, 2010
Additional Notes: This work grew out of a stay of all three authors at the Fields Institute in Toronto, Canada during the Thematic Program on Operator Algebras in the fall of 2007. They would like to acknowledge the pleasant stay and the very inspiring atmosphere at the Fields Institute.
The second author’s research was supported by NSF Grant DMS-0701162
The third author’s research was supported by Discovery and LSI grants from NSERC (Canada) and by the Killam Fellowship from the Canada Council for the Arts
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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