Resolvents of $\mathscr {R}$-diagonal operators
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- by Uffe Haagerup, Todd Kemp and Roland Speicher PDF
- Trans. Amer. Math. Soc. 362 (2010), 6029-6064 Request permission
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 $|\lambda -c|^2$ where $c$ is Voiculescu’s circular operator, and we give an asymptotic formula for the negative moments of $|\lambda -a|^2$ 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.References
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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
- MR Author ID: 771033
- Email: tkemp@math.ucsd.edu
- Roland Speicher
- Affiliation: Department of Mathematics and Statistics, Queen’s University, Jeffery Hall, Kingston, Ontario, Canada K7L 3N6
- Email: speicher@mast.queensu.ca
- 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 - © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 6029-6064
- MSC (2010): Primary 46L54
- DOI: https://doi.org/10.1090/S0002-9947-2010-05140-2
- MathSciNet review: 2661507