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Anal. 23 (1992), 1623-1638. H. BERCOVICI & C. edu Received: August 23rd, 1995. A. edu Operator Theory: Advances and Applications, Vol. 104 © 1998 Birkhliuser Verlag Basel/Switzerland Regularity Questions for Free Convolution HARI BERCOVICI AND DAN VOICULESCU* For Carl Pearcy on his sixtieth birthday 1. INTRODUCTION Free additive convolution is a binary operation on the set M of all probability measures on the real line R. This operation was first defined in [7] for measures with finite moments of all orders (in particular for compactly supported measures).

Then we have for every y > 0 and a E (0,2). Iflimy---+oow(iy)jy for a = 2 as well. 4. 1 in [3]). 1. L, v E M, there exists an analytic function w : C+ C+ such that (i) GJLEEv = GJL 0 w; -t Regularity Questions for Free Convolution (ii) ~w(z) ~ ~z for Z E C+; and (iii) w(z) = z(l + 0(1)) when z ----; 00 41 nontangentially to R. We will not prove here the existence of w for which we refer to Biane [3]. We will just argue that (ii) and (iii) are true. Recall first that w must have a Nevanlina representation, say w(z) = a +,Bz + 1 CXl -CXl 1 + tz - - du(t), t- z where a E R, ,B ~ 0, and u is a finite positive measure on R.

LARSON, Reflexive algebras with finite width lattices: Tensor products, cohomology, compact perturbations, J. Flmct. Analysis 55 (1984), 176-199. D. HADWIN, Algebraically reflexive linear transformations, Linear and Multilinear Algebra 14 (1983), 225-233. D. HADWIN AND E. A. NORDGREN, Reflexivity and direct sums, Acta Sci. Math. 55 (1991), 181-197. 1. KAPLANSKY, Infinite Abelian Groups. University of Michigan Press, Ann Arbor, 1969. A. KATAVOLOS, M. S. LAMBROU, AND M. PAPADAKIS, On some algebras diagonalIntegral Equations Operator Theory 17 (1993), 68-94.