Resolvents of R-diagonal operators.

TY - JOUR

T1 - Resolvents of R-diagonal operators.

AU - Haagerup, Uffe

AU - Kemp, Todd

AU - Speicher, Roland

PY - 2010/11

Y1 - 2010/11

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 -