Resolvents of R-diagonal operators.

Uffe Haagerup; Todd Kemp; Roland Speicher

Resolvents of R-diagonal operators.

Uffe Haagerup, Todd Kemp, Roland Speicher

Department of Mathematical Sciences

1 Citation (Scopus)

Abstract

We consider the resolvent (λ-a)^-1 of any R-diagonal operator a in a II₁-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 R-transform of the operator |λ-c|² where c is Voiculescu's circular operator, and we give an asymptotic formula for the negative moments of |λ - a|² for any 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.

Original language	English
Journal	Transactions of the American Mathematical Society
Volume	362
Pages (from-to)	6029-6064
ISSN	0002-9947
Publication status	Published - Nov 2010

Cite this

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abstract = "We consider the resolvent (λ-a)-1 of any R-diagonal operator a in a II1-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 R-transform of the operator |λ-c|2 where c is Voiculescu's circular operator, and we give an asymptotic formula for the negative moments of |λ - a|2 for any 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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AU - Haagerup, Uffe

AU - Kemp, Todd

AU - Speicher, Roland

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N2 - We consider the resolvent (λ-a)-1 of any R-diagonal operator a in a II1-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 R-transform of the operator |λ-c|2 where c is Voiculescu's circular operator, and we give an asymptotic formula for the negative moments of |λ - a|2 for any 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.

AB - We consider the resolvent (λ-a)-1 of any R-diagonal operator a in a II1-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 R-transform of the operator |λ-c|2 where c is Voiculescu's circular operator, and we give an asymptotic formula for the negative moments of |λ - a|2 for any 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.

M3 - Journal article

SN - 0002-9947

VL - 362

SP - 6029

EP - 6064

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

ER -

Resolvents of R-diagonal operators.

Abstract

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