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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
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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Additional Information

Uffe Haagerup
Affiliation: Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55 DK-5230 Odense M, Denmark

Todd Kemp
Affiliation: Department of Mathematics, University of California, San Diego, La Jolla, California 92093

Roland Speicher
Affiliation: Department of Mathematics and Statistics, Queen’s University, Jeffery Hall, Kingston, Ontario, Canada K7L3N6

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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