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Regularity questions for free convolution. (English) Zbl 0927.46048
Bercovici, Hari (ed.) et al., Nonselfadjoint operator algebras, operator theory, and related topics. The Carl M. Pearcy anniversary volume on the occasion of his 60th birthday. Basel: Birkhäuser. Oper. Theory, Adv. Appl. 104, 37-47 (1998).
The authors study, whether regularity properties of probability measures \( \mu \), \( \nu \) on \( \mathbb R \) are inherited by the free additive convolution \( \mu\boxplus\nu \). Having established in [Indiana Univ. Math. J. 42, No. 3, 733-773 (1993; Zbl 0806.46070)] that the operation \( \boxplus \) introduced by D. Voiculescu [Lect. Notes Math. 1132, 556-588 (1985; Zbl 0618.46048)] extends to arbitrary probability measures, they generalize some results known in the case of compactly supported \( \mu \) and \( \nu \). In terms of the Cauchy transform defined by \( G_\mu(z)=\int_{-\infty}^{+\infty}(z-t)^{-1} d\mu(t) \), \( z=x+iy \), \( y>0 \), \( G_{\mu\boxplus\nu} \) is subordinated to \( G_\mu \) in \( \mathbb C^+ \), i.e. there exists an analytic function \( \omega :\mathbb C^+\to\mathbb C^+ \) such that 1) \( G_{\mu\boxplus\nu}=G_\mu\circ \omega \), 2) \( \mathop{\text{Im}}\omega(z)\geqslant\mathop{\text{Im}}z \), \( z\in\mathbb C^+ \) and 3) \( \omega(z)=z(1+o(1)) \), \( z\to\infty \) nontangentially to \( \mathbb R \). Then, for any fixed \( y>0 \) and \( 1<p\leqslant\infty \), \( \| (G_{\mu\boxplus\nu})_y\| _p\leqslant\| (G_\mu)_y\| _p \) in \( L^p(\mathbb R) \), and similar relations hold for the real (resp. imaginary) parts of the transforms. If \( \mu \) is absolutely continuous with density \( f\in L^p(\mathbb R) \), so is \( \mu\boxplus\nu \) and its density \( g \) satisfies \( \| g\| _p\leqslant\| f\| _p \). Some inequalities involving the Riesz energies are obtained and all the above are symmetrical in \( \mu \) and \( \nu \). If the distribution function \( {\mathcal F}_\mu \) of \( \mu \) is \( \alpha \)-Hölder, \( \alpha\in(0,1] \), (with constant \( c \)), the same is true of \( {\mathcal F}_{\mu\boxplus\nu} \) (with constant \( \leqslant c \)). But there exist compactly supported \( \mu \), \( \nu \) having densities of class \( C^\infty \) and \( C^1 \), respectively, such that the density of \( \mu\boxplus\nu \) is not of class \( C^1 \). A description of atoms is given: \(\gamma\in\mathbb R\) is an atom of \(\mu\boxplus\nu\iff\) there exist atoms \(\alpha,\beta\) for \(\mu,\nu\), respectively, with \(\gamma=\alpha+\beta\) and \((\mu\boxplus\nu)(\{\gamma\})=\mu(\{\alpha\})+\nu(\{\beta\})-1>0\). The background of noncommutative free random variables is mentioned occasionally.
For the entire collection see [Zbl 0892.00040].

MSC:
46L54 Free probability and free operator algebras
46L53 Noncommutative probability and statistics
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