Partially defined semigroups relative to multiplicative free convolution
Nica and Speicher have shown that every probability measure on the line belongs to a partial semigroup {μt : t ≥ 1} relative to additive free convolution (i.e., μt+s=μt⊞μs for t, s≥1). We prove analogous results for multiplicative free convolution on the positive half-line and on the circle. The exi...
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| Published in | International Mathematics Research Notices Vol. 2005; no. 2; pp. 65 - 101 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Hindawi Publishing Corporation
01.01.2005
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| Online Access | Get full text |
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| Abstract | Nica and Speicher have shown that every probability measure on the line belongs to a partial semigroup {μt : t ≥ 1} relative to additive free convolution (i.e., μt+s=μt⊞μs for t, s≥1). We prove analogous results for multiplicative free convolution on the positive half-line and on the circle. The existence of semigroups is derived from certain global inversion results for analytic functions defined on the disk or on the slit complex plane. A close analysis of the global inverses yields regularity results for the measures in these semigroups. We also use this analysis to improve the known results for the additive situation. |
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| AbstractList | Nica and Speicher have shown that every probability measure on the line belongs to a partial semigroup {μt : t ≥ 1} relative to additive free convolution (i.e., μt+s=μt⊞μs for t, s≥1). We prove analogous results for multiplicative free convolution on the positive half-line and on the circle. The existence of semigroups is derived from certain global inversion results for analytic functions defined on the disk or on the slit complex plane. A close analysis of the global inverses yields regularity results for the measures in these semigroups. We also use this analysis to improve the known results for the additive situation. |
| Author | Belinschi, S. T. Bercovici, H. |
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