Powers of R-Diagonal Elements

J. OPERATOR THEORY c Copyright by Theta, 2002 47(2002), 197–212

POWERS OF R-DIAGONAL ELEMENTS

FLEMMING LARSEN

Communicated by William B. Arveson

Abstract. We prove that if (a, b) is an R-diagonal pair in some non-commutative probability space (A, ϕ) then (ap, bp) is R-diagonal too and we compute the determining series f(ap,bp) in terms of the distribution of ab. We give estimates of the upper and lower bounds of the support of free multiplicative convolution of probability measures compactly supported on [0, ∞[, and use the results to give norm estimates of powers of R-diagonal elements in ﬁnite von Neumann algebras. Finally we compute norms, distributions and R- transforms related to powers of the circular element.

Keywords: Free probability theory, R-diagonal elements, operator algebras.

MSC (2000): 46L54.

1. INTRODUCTION AND PRELIMINARIES

In the setup of Free Probability Theory we study certain random variables. By a non-commutative probability space (A, ϕ) we mean a unital algebra A (over the complex numbers) equipped with a unital functional ϕ. If A is a von Neumann algebra and ϕ is normal we call (A, ϕ) a non-commutative W ∗-probability space. We write ( , τ) for a non-commutative W ∗-probability space with a faithful normal M tracial state τ. Elements in A are called random variables, and the distribution µa of a random variable in (A, ϕ) is the linear functional µ : C[X] C determined a → by µa(P ) = ϕ(P (a)) for all P in C[X]. We refer to [16] for the basic facts of Free Probability Theory and record here for easy reference what we need in this paper. 198 Flemming Larsen

If a is a self-adjoint element in a non-commutative W ∗-probability space there exists a unique compactly supported probability measure (also denoted µa) such that p p ϕ(a ) = t dµa(t), p N. Z ∈ R In this case supp µ sp a. We often view these measures as distributions in the a ⊆ sense of [16]. In the following we introduce the R- and S-transforms of distributions, and the definitions carry over to measures. The S-transform of a distribution with non-vanishing first moment is Sµ defined as a formal power series in the following way (cf. [16]): define the moment series ψµ as ∞ n n ψµ(z) = µ(X )z nX=1 and let χµ denote the unique inverse formal power series (with respect to composition) of ψµ. This series is of the form

∞ n χµ(z) = αnz nX=1 where α = µ(X)−1 = 0. Then we deﬁne 1 6 ∞ z + 1 (z) = χ (z) = (z + 1) α zn−1 Sµ z µ n nX=1 as a formal power series. (Note that (0) = µ(X)−1.) The S-transform converts Sµ multiplicative free convolution into multiplication of formal power series in the following way: = whenever µ and ν are distributions with non- Sµ ν Sµ · Sν vanishing ﬁrst moments. If µ is a compactly supported probability measure on [0, [ we let m(µ) = ∞ min supp µ, r = r(µ) = max supp µ and we can view µ as the distribution of a positive element in a suitably chosen non-commutative von Neumann probability space. We have then zs ψµ(z) = dµ(s) Z 1 zs R −

−1 hence ψµ is analytic on z C z / supp µ , and χµ, µ are analytic in a { ∈ | ∈ } S p neighbourhood of ψµ ([0, 1/r[), cf. [4]. We denote the moments of µ by µ(X ) = tp dµ(t) for every natural number p. R Powers of R-diagonal elements 199

Pringsheim’s theorem shows that 1/r is a non-removable singularity for ψµ and the behaviour of ψµ near 1/r can be classified into one of the following three cases: (i) ψ is unbounded near 1/r : ψ (t) as t 1/r ; µ µ → ∞ → − (ii) ψ is bounded and ψ0 is unbounded near 1/r: ψ0 (t) as t 1/r µ µ µ → ∞ → − and lim ψµ(t) exists and is finite; t→1/r− 0 0 (iii) ψµ and ψµ are bounded near 1/r: lim ψµ(t) and lim ψµ(t) exist t→1/r− t→1/r− and are finite.

This makes it possible to determine r in terms of the function χµ: (i) if χ is analytic in a neighbourhood of [0, [ and χ0 > 0 on [0, [ then µ ∞ µ ∞ 1/r = lim χµ(y); y→∞ 0 (ii) if χµ is analytic in a neighbourhood of [0, y0[, χµ > 0 on [0, y0[ and y0 is the largest number with these properties then 1/r = lim χµ(y). y→y0−

Since χµ is increasing we can estimate r : 1/r > lim χµ(y) whenever χµ is y→y0− 0 analytic on a neighbourhood of [0, y0[ and χµ > 0 on ]0, y0[. By V (µ) we denote the variance of the measure µ : V (µ) = µ(X 2) µ(X)2, − and if µ(X) > 0 we can bound r(µ) from below:

µ(X2) = x2 dµ(x) 6 r(µ)µ(X) Z R hence V (µ)/µ(X) + µ(X) 6 r(µ). For a measure µ we let µ−1 denote the image measure of µ induced by the reciprocal map x 1/x, and let µ denote the image measure induced by 7→ sq the squaring function sq : z z2. Note that if µ is supported on ]0, [ then 7→ ∞ r(µ−1) = m(µ)−1. The R-transform of a distribution µ was introduced by Voiculescu in [14] Rµ (see also [16]) as a formal power series obtained in the following way: Deﬁne

∞ n −n−1 Gµ(z) = µ(X )z nX=0 as a formal Laurent series. (The symbol Gµ will be referred to as the Cauchy transform of µ.) Then Gµ is invertible with respect to composition and the inverse −1 Gµ is of the form ∞ 1 G−1(z) = + α zn µ z n nX=0 200 Flemming Larsen and is deﬁned to be the power series part of G−1: Rµ µ ∞ (z) = α zn. Rµ n nX=0 The R-transform converts additive free convolution of distributions into addition of formal power series: = + for all distributions µ and ν. Rµ ν Rµ Rν In [7] the (1-dimensional) R-transform was generalized to multidimensional distributions µ : C X i I C and the R-transform of µ is then denoted R . h i | ∈ i → µ In the 1-dimensional case we have the relation R (z) = z (z). µ Rµ The circular element c (of norm 2) was introduced in [15] and the polar decomposition c = uh was determined: u and h are -free, u is a Haar unitary ∗ (every non-trivial moment of u is 0) and h is quarter circular (of radius 2):

1 2 dµh = 4 x 1[0,2](x) dx. π p − · A model for the circular element is the following: Let be a Hilbert space with ∞ H orthonormal basis ξ , ξ , let ( ) = C Ω ⊗n be the full Fock space { 1 2} T H ⊕ H nL=1 of and let l1, l2 be the creation operators of ξ1 and ξ2 respectively. Then H ∗ ∗ c = (l1 +l2)/√2 is a circular element in the ﬁnite non-commutative W -probability space (W ∗(c, c∗), Ω, Ω ). h · i The R-transforms of the -distributions of c and u have similar forms, cf. [10]: ∗

(1.1) Rµ(c,c∗) (z1, z2) = z1z2 + z2z1, ∞ ∞ (1.2) R (z , z ) = ( 1)n−1C (z z )n + ( 1)n−1C (z z )n. µ(u,u∗) 1 2 − n−1 1 2 − n−1 2 1 nX=1 nX=1

(p) pn The numbers C are the Catalan numbers (cf. [2] and [6]): Cn = /n n n 1 − (2) 2n is the n’th Fuss-Catalan number of parameter p and C = Cn = /n, n n 1 n, p N. For convenience we let C = 1. In [10] Nica and Speicher (see−also [8] ∈ 0 for a more general deﬁnition) introduced the class of R-diagonal pairs in non- commutative probability spaces as those pairs (a, b) whose R-transform is of the form ∞ ∞ n n Rµ(a,b) (z1, z2) = αn(z1z2) + αn(z2z1) nX=1 nX=1 ∞ where α C. The series f (z) = α zn is called the determining se- n ∈ µ(a,b) n nP=1 ries of Rµ(a,b) . An R-diagonal element is an element a (in a non-commutative -probability space) such that (a, a∗) is an R-diagonal pair. ∗ Powers of R-diagonal elements 201

The determining series fµ(a,b) can be obtained from the moment series ψµab using the -operation on formal power series, cf. [10]. In dimension 1 the oper- ∗ ation satisfies (and can be defined by) R = R R for arbitrary free ∗ µab µa ∗ µb random variables a and b in some non-commutative probability space. Then R = ψ Möb where Möb is the series µa µa ∗ ∞ Möb(z) = ( 1)n−1C zn, − n−1 nX=1

and fµ(a,b) = Rµab Möb. (If a and b are random variables in a tracial non- ∗ ∞ n commutative probability space then Rµab = Rµba .) The series ζ(z) = z is the n=1 inverse to Möb with respect to : Möb ζ(z) = ζ Möb(z) = z. P ∗ ∗ ∗ In the n-dimensional case we let Mµ denote the moment series of an n- dimensional distribution µ : C X , . . . , X C: h 1 ni → ∞ n M (z , . . . , z ) = µ(X X )z z . µ 1 n i1 · · · ik i1 · · · ik kX=1 i1,...X,ik=1

Then R = M Möb , M = R ζ , where Möb and ζ are n-dimensional µ µ ∗ n µ µ ∗ n n n analogues of Möb and ζ respectively, and likewise for , cf. [10]. ∗ We often simplify the notation and write R(a,b) in place of Rµ(a,b) etc. If a is a random variable in a non-commutative C∗-probability space the

R-transform of µa is analytic in a neighbourhood of 0, cf. [4]. The paper is organized as follows. In Section 2 we show that any power of any R-diagonal element is R-diagonal. In Section 3 we derive some estimates on the radius of the support of the free multiplicative convolution power of a measure compactly supported in [0, [. In Section 4 we compute distributions and R-series ∞ related to powers of the circular element.

2. POWERS OF R-DIAGONAL PAIRS

In the sequel we let M 0(R) denote the set of symmetric compactly supported probability measures on R, and let rj (µ) denote the j’th coeﬃcient in Rµ:

∞ j Rµ(z) = rj (µ)z . Xj=1

Proposition 5.2 in [11] shows that if µ M 0(R) then r (µ) = 0 for all j in N. ∈ 2j−1 202 Flemming Larsen

Lemma 2.1. The sets

(µ(X2), . . . , µ(X2n)) µ M 0(R) , (r (µ), . . . , r (µ)) µ M 0(R) ∈ 2 2n ∈ have interior points for all natural numbers n.

1 Proof. The measure µ = 2 1[−1,1](x) dx is symmetric and the even moments are µ(X2k) = (2k + 1)−1 (k N). Fix a natural number n and put ∈ n n B1 = (α1, . . . , αn) R αj < 1 . n ∈ | | o Xj=1

Let P2j denote the normalized Legendre polynomial of order 2j, and for (α1, . . . , αn) B we deﬁne ∈ 1 1 dµ = P (x) + α P (x) + + α P (x) 1 (x) dx. (α1,...,αn) 2 0 1 2 · · · n 2n · [−1,1] Observe that P is an even polynomial, P (x) 6 1 for x 6 1 hence 2j | 2j | | | n αj P2j (x) 6 1

Xj=1 on [ 1, 1]. The orthogonality properties of the sequence (P ) N implies that − 2j j∈ 1 1 2i 2j x P2j (x) dx = 0, x P2j (x) dx = 0 Z Z 6 −1 −1 whenever i = 0, . . . , j 1, j N. In particular it follows that µ M 0(R) − ∈ (α1,...,αn) ∈ (for (α , . . . , α ) B ). Then 1 n ∈ 1

n 1 n 2k 1 1 2k 2k µ (X ) = + αj x P2j (x) dx = µ(X ) + bkj αj , (α1,...,αn) 2k + 1 2 Z Xj=1 −1 Xj=1

1 where b = 1 x2kP (x) dx. Especially b = 0 (j = 1, . . . , n) and b = 0 if kj 2 2j jj 6 kj −R1 k < j. Then

2 2 µ(α1,...,αn)(X ) µ(X ) b11 0 α1 α1 . . . ·.· · . . . (2.1)  .   . = . .. .  . =B  .  . − . 2n 2n  µ (X )   µ(X )   bn1 bnn  αn   αn   (α1,...,αn)     · · ·     Powers of R-diagonal elements 203

n 2 where B = (bij )i,j=1. Since B is invertible Equation (2.1) shows that (µ(X ), . . . , µ(X2n)) is an interior point in (ν(X 2), . . . , ν(X2n)) ν M 0(R) . { | ∈ } There exist (universal) continuous functions F, G : Cn Cn such that → 2 2 r2(ν) ν(X ) ν(X ) r2(ν) . . . .  .  = F  .  ,  .  = G  .  , ν M 0(R) . . . . ∈  r (ν)   ν(X2n)   ν(X2n)   r (ν)   2n       2n  and F G = idCn = G F . In particular F is open and (r (ν), . . . , r (ν)) ν ◦ ◦ { 2 2n | ∈ M 0(R) has an interior point. } ∞ Remark. Let ( , τ) = L(Z2) (L (µ), dµ). Then is a ﬁnite M ∗ µ∈M 0(R) · M von Neumann algebra with a faithful normal trace τR. Let a be a generating ∗ unitary in L(Z2). If T is an arbitrary element in some ﬁnite non-commutative W - probability space it follows from Corollary 3.2 in [5] that the element a id ∞ · L (µ˜|T |) is an R-diagonal element in and that it has the same -distribution as T . Thus M ∗ contains a representative of every R-diagonal element. Let M ∞ ∞ j j Sn = (α1, . . . , αn) T : R(T,T ∗)(z1, z2) = αj (z1z2) + αj (z2z1) . ∃ ∈ M Xj=1 Xj=1

It follows from Proposition 5.2 in [11] that

S = (r (µ), . . . , r (µ)) µ M 0(R) , n { 2 2n | ∈ } thus Sn has an interior point according to Lemma 2.1.

Theorem 2.2. Let (a, b) be an R-diagonal pair in a non-commutative probability space (A, ϕ), and let p N. Then (ap, bp) is an R-diagonal pair with ∈ determining series ∗ p f p p = R Möb. (a ,b ) ab ∗ ∗ p In particular Rapbp = Rab . Proof. Let n N, i , . . . , i 1, 2 . We first assume that (i , . . . , i ) de- ∈ 1 n ∈ { } 1 n notes an index off the diagonal, i.e., (i , . . . , i ) = (1, 2, . . . , 1, 2) and (i , . . . , i ) = 1 n 6 1 n 6 (2, 1, . . . , 2, 1). We note the existence of a universal polynomial P such that

[coef(i1, . . . , in)]R(ap,bp) = P [coef(1, 2)]R(a,b), . . . , [coef(1, 2, . . . , 1, 2)]R(a,b) 2pn | {z } 204 Flemming Larsen whenever (a, b) is an R-diagonal pair in some non-commutative probability space. Indeed, if R C z , z is of the form ∈ h 1 2i ∞ ∞ j j R(z1, z2) = αj (z1z2) + αj (z2z1) Xj=1 Xj=1 we deﬁne M, M (p), R(p) C z , z by ∈ h 1 2i

M = R ζ , M (p)(z , z ) = M(zp, zp), R(p) = M (p) M¨ob , ∗ 2 1 2 1 2 ∗ 2

(p) and it follows that there exists a polynomial P such that [coef(i1, . . . , in)]R =

P (α1, . . . , αpn). If a and b are random variables in a non-commutative probability space (A, ϕ) and (a, b) is an R-diagonal pair with determining series f(a,b)(z) = ∞ j αj z , then jP=1 [coef(i1, . . . , in)]R(ap,bp) = P (α1, . . . , αpn).

If T is an R-diagonal element in ( , τ) it follows from Proposition 3.10 M p in [5] that T is R-diagonal. This implies that P (α1, . . . , αpn) = 0 whenever (α , . . . , α ) S . Since S has an interior point this implies that P = 0 hence 1 pn ∈ pn pn [coef(i1, . . . , in)]R(ap,bp) = 0.

We next assume that (i1, . . . , in) = (1, 2, . . . , 1, 2). Then a symmetry argu- n ment reveals that | {z }

[coef(1, 2, . . . , 1, 2)]R(ap,bp) = [coef(2, 1, . . . , 2, 1)]R(ap,bp). n n | {z } | {z } p p Since n and i1, . . . , in are arbitrary we conclude that (a , b ) is R-diagonal. p p Finally we compute the determining series f(ap,bp) for (a , b ). Note that for given n there exists a universal polynomial P such that

∗ p [coef(n)]f p p [coef(n)]R M¨ob = P (α , . . . , α ) (a ,b ) − ab ∗ 1 pn

∗ where α = [coef(j)]f . If (T, T ) is an R-diagonal pair in ( , τ) then f p p ∗ j (a,b) M (T ,(T ) ) = R p p ∗ Möb = R Möb = R ∗ p Möb (cf. Proposition 3.10 in [5]) T (T ) p µ ∗ µT T ∗ ∗ T T ∗ ∗ ∗ p whence P (S ) = 0 . We conclude that P = 0 and thus f p p = R Möb. pn { } (a ,b ) ab ∗ Powers of R-diagonal elements 205

3. NORM-ESTIMATES OF POWERS AND PRODUCTS OF R-DIAGONAL ELEMENTS

Theorem 3.1. Let p be a natural number, let µ, µ1, . . . , µp be compactly supported probability measures on [0, [. Then: ∞ (i) r(µ p) 6 ep r(µ)µ(X)p−1;

(ii) if r(µj ) > 0 for all j = 1, . . . , p then

r(µj ) r(µ1 µp) 6 ep max µ1(X) µp(X); · · · j=1,...,p µj (X) · · · ·

(iii) r(µ p) > µ(X)p + pV (µ)µ(X)p−2;

(iv) if r(µj ) > 0 for all j = 1, . . . , p then

p V (µ ) r(µ µ ) > µ (X) µ (X) 1 + j . 1 · · · p 1 · · · p µ (X)2 Xj=1 j

Proof. The idea of the proof of (i) is to ﬁnd an interval ]0, y0[ on which χµ p 0 p −1 is analytic and χ p > 0. Then r(µ ) > lim χµ p (y). µ y→y0− The statement holds trivially for p = 1 and for µ = δ0 so let p be a natural number greater than 2 and assume that µ = δ . Then 6 0 p−1 z + 1 p (3.1) χµ p (z) = χµ(z) z and it follows that χµ p is analytic in a neighbourhood of ]0, y0[ if χµ is analytic 0 in a neighbourhood of ]0, y0[. It follows from (3.1) that χµ p > 0 on ]0, y0[ if and only if p 1 χ0 (y) − < y(1 + y) µ p χµ(y)

0 for all y in ]0, y0[. Inserting y = ψµ(t) we infer that χµ p > 0 on ]0, y0[ if and only if

p 1 ψµ(t)(1 + ψµ(t)) (3.2) − < 0 p tψµ(t) for all t in ]0, t0[ where t0 = lim χµ(y). y→y0− −1 Choose t0 = (p r(µ)) . Using the integral formula for ψµ we estimate:

ψ (t)(1 + ψ (t)) ψ (t) µ µ µ > 0 > ts 1 tr(µ) tψµ(t) (1−ts)2 dµ(s) − R 206 Flemming Larsen hence (3.2) holds for all t in ]0, t0[. Then ψµ is analytic in a neighbourhood of ]0, t0] hence χµ is analytic in a neighbourhood of [0, y0] and we can estimate: ψ (t ) s µ(X) µ 0 = dµ(s) 6 , t0 Z 1 st0 1 r(µ)t0 R − − p−1 p−1 1 r(µ)t0 1 −1 p−1 > > χµ p (ψµ(t0)) (ψµ(t0) + 1) − t0 p−1 (1 p ) µ(X) p r(µ)µ(X) − whence

1 p−1 r(µ p) 6 p r(µ)µ(X)p−1 1 + 6 ep r(µ)µ(X)p−1, p 1 − which shows (i). Put α = (µ (X) µ (X))1/p and let ν (j = 1, . . . , p) be the image measure 1 · · · p j (µ ) . Then ν ν = µ µ and ν (X) = = ν (X) = α. j z7→αz/µj (X) 1 · · · p 1 · · · p 1 · · · p Due to the foregoing analysis we have that 0 p 1 χν (y) (3.3) − < y(y + 1) j p χνj (y)

(j) (j) −1 for all y in ]0, y0 [ where y0 = ψνj (p r(νj )) . Put r = max r(νj ) and j=1,...,p −1 −1 t0 = min (p r(νj )) = (p r) . Then (3.3) holds on ]0, y0,j[ where y0,j = j=1,...,p

ψνj (t0) hence the estimate (3.3) holds on ]0, y0[ where y0 = min y0,j. We j=1,...,p assume without loss of generality that y0 = y0,1. Note that χνj (y0) 6 χνj (y0,j ) =

χνj (ψνj (t0)) = t0. Then 0 1 χν (y) (p 1) < p min j − y(y + 1) j=1,...,p χνj (y) for all y in ]0, y0[ and we conclude that

p 0 d χνj (y) 1 log χ (y) = (p 1) > 0 dy ν1 ··· νp χ (y) − − y(y + 1) Xj=1 νj for all y in ]0, y [. Thus χ0 > 0 on ]0, y [, and since p > 2 each χ is 0 ν1 ··· νp 0 νj analytic on a neighbourhood of ]0, y0], hence y + 1 p−1 −1 > 0 r(ν1 νp) χν1 ··· νp (y0) = χν1 (y0) χνp (y0) · · · y0 · · · p p χν (y0) χν (y0) > χ (ψ (t )) j = t j ν1 ν1 0 y 0 ψ (χ (y )) jY=2 0 jY=2 νj νj 0 p p p 1 r(νj )χν (y0) 1 rt 1 p−1 1 > t − j > t − 0 = t 1 . 0 ν (X) 0 ν (X) 0 − p ν (X) jY=2 j jY=2 j jY=2 j Powers of R-diagonal elements 207

Therefore 1 p−1 r(µ µ ) = r(ν ν ) 6 pr 1 αp−1 1 · · · p 1 · · · p − p 1 − αr(µj ) p−1 r(µj ) 6 ep max α = ep max µ1(X) µp(X) j=1,...,p µj (X) j=1,...,p µj (X) · · · · and this proves (ii). To prove (iv) we ﬁrst note that if µ is a measure with non-vanishing ﬁrst moment the power series of is Sµ 1 V (µ) (z) = z + O(z2) Sµ µ(X) − µ(X)3 as z 0. This implies that → p 1 V (µj ) 2 (z) = 1 z + O(z ) Sµ1 ··· µp µ (X) − µ (X)2 jY=1 j j p 1 V (µ ) = 1 j z + O(z2) µ (X) µ (X) − µ (X)2 1 · · · p Xj=1 j hence p V (µ µ ) 1 V (µ ) 1 · · · p = j , µ µ (X)3 µ (X) µ (X) µ (X)2 1 · · · p 1 · · · p Xj=1 j and thus

V (µ1 µp) r(µ1 µp) > · · · + µ1 µp(X) · · · µ1 µp(X) · · · · · · p V (µ ) = µ (X) µ (X) 1 + j . 1 · · · p µ (X)2 Xj=1 j

This shows (iv). If µ is a Dirac measure then r(µ p) = µ(X)p, V (µ) = 0 and (iii) is fulﬁlled. If µ is not a Dirac measure then (iii) follows from (iv). Corollary 3.2. Let p N, T, T , . . . , T be R-diagonal elements in a non- ∈ 1 p commutative probability space ( , τ) with a faithful normal tracial state. Then M T p 6 √ep T T p−1 k k k k k k2 for every natural number p, and if T , . . . , T are -free and T , . . . , T = 0 then 1 p ∗ 1 p 6

Tj T1 Tp 6 √ep max k k T1 2 Tp 2. k · · · k j=1,...,p T · k k · · · k k k jk2 208 Flemming Larsen

p Proof. It follows from Propositions 3.6 and 3.10 in [5] that T and T1 Tp p · · · are R-diagonal and that µ p 2 = µ , µ 2 = µ 2 µ 2 . Then we |T | |T |2 |T1···Tp| |T1| · · · |Tp| estimate:

2 r(µ|Tj | ) 2 2 6 2 2 r(µ|T1 | µ|Tp| ) ep max µ|T1| (X) µ|Tp| (X) j=1,...,p 2 · · · µ|Tj | (X) · · · · 2 Tj 2 2 = ep max k k T1 2 Tp 2 j=1,...,p T 2 · k k · · · k k k jk2 and the conclusion follows. If µ is not a Dirac measure Theorem 3.1 (i) shows that r(µ p) = O(pµ(X)p) and (iii) shows that this is the best asymptotic estimate. Thus T p = O(√p T p) k k k k2 is the best asymptotic estimate for the norm of an R-diagonal element (compare to the estimate T p 6 (1 + p) T T p−1 obtained in Corollary 4.2 in [5]). k k k k k k2 Corollary 3.3. If p is a natural number, and µ, µ1, . . . , µp are compactly supported probability measures on ]0, [, then ∞ −1 min m(µj )µ (X) 1 j=1,...,p j > m(µ1 µp) −1 −1 , · · · ep · µ (X) µp (X) (3.4) 1 · · · 1 m(µ) m(µ p) > . ep · (µ−1(X))p−1

p ∗ ∞ Proof. In the ﬁnite non-commutative W -probability space L (µj ), j=1 dµ we can ﬁnd a free family a , . . . , a of positive invertible elements such · j { 1 p} Rthat the distribution (as a measure) of aj is µj for all j. Then the distribution of a a is µ µ and the distribution of (a a )−1 is 1 · · · p 1 · · · p 1 · · · p −1 −1 −1 −1 −1 −1 −1 µ(a1···ap) = µa ···a = µa µa = µ1 µp . p 1 p · · · 1 · · · Then

m(µ µ )−1 = r (µ µ )−1 = r(µ−1 µ−1) 1 · · · p 1 · · · p 1 · · · p −1 r(µj ) −1 −1 6 ep max µ1 (X) µp (X) j=1,...,p −1 µj (X) · · · · hence −1 min m(µj )µ (X) 1 j=1,...,p j > m(µ1 µp) −1 −1 · · · ep · µ (X) µp (X) 1 · · · and the conclusion follows. Powers of R-diagonal elements 209

Let µ be a compactly supported probability measure on ]0, [. For the free ∞ multiplicative convolution power µ p of µ we have the obvious estimate

(3.5) m(µ p) > m(µ)p.

If µ is a Dirac measure then m(µ p) = m(µ)p. In the following we assume that µ is not a Dirac measure. Then

m(µ)µ−1(X) = m(µ) t−1 dµ(t) < m(µ) m(µ)−1 dµ(t) = 1 Z Z R R and the estimate

p m(µ) p−1 = ep m(µ)µ−1(X) 0 as p m(µ) → → ∞ epµ−1(X) shows that the estimate (3.4) is sharper than (3.5) for large p.

4. POWERS OF THE CIRCULAR ELEMENT

In this section we compute the norm of every power of the circular element c. Fur- thermore we determine the distribution of cp 2 and the R-transform of (cp, (cp)∗) | | for any p in N. It is recently shown in [12] that powers of the circular element are R-diagonal elements, and the coeﬃcients in the R-transforms were computed. Proposition 4.1. Let c be the circular element and let p be a natural number. Then: (i) cp 2 = (p + 1)p+1/pp; k k p 2 n (p+1) (ii) the moments of c are µ p 2 (X ) = Cn for all natural numbers p | | |c | and n; p (iii) the determining function f(cp,(cp)∗) for the R-diagonal element c is

∞ (p−1) n f(cp,(cp)∗)(z) = Cn z . nX=1 for every natural number p greater than 2.

p 2 p Proof. Let µ = µ|c|2 . We ﬁrst note that c = r(µ|cp|2 ) = r(µ ). It is k k 2 shown in [5] that (z) = 1/(z+1) hence χ (z) = z/(z+1) and χ p (z) = z/(z+ Sµ µ µ 1)p+1 are analytic in a neighbourhood of [0, [. It is straightforward to verify 0 −1 0 −1 ∞ p −1 that χµ p > 0 on ]0, p [ and χµ p (p ) = 0 hence 1/r(µ ) = χµ p (p ) = pp/(p + 1)p+1 and (i) follows. 210 Flemming Larsen

For n, p natural numbers we ﬁnd

n n µ p 2 (X ) = the coeﬃcient of z in ψ (z) |c | µ|cp|2

(∗) 1 −n 1 −n = Res(χµ (z) , z = 0) = Res(χ p (z) , z = 0) n |cp|2 n µ 1 (z + 1)(p+1)n 1 (p + 1)n = Res , z = 0 = = C(p+1). n zn n n 1 n − At ( ) we use Lagranges Inversion Theorem, cf. Section 3.8 in [1], [3]. This ∗ shows (ii). ∞ To prove (iii) we ﬁrst note that (z) = (1 z)−1 hence R (z) = zj = Rµ − µ jP=1 ζ(z). Then

∗ p R p p = R p p = R p = R R = ζ µc (c )∗ µ(c )∗c µ µ ∗ · · · ∗ µ p terms | {z } hence ∗ p ∗ (p−1) f p p ∗ = R p p Möb = ζ Möb = ζ . (c ,(c ) ) µc (c )∗ ∗ ∗ ∞ n If p = 2 then f(cp,(cp)∗)(z) = ζ(z) = z and if p > 2 then nP=1 ∗ (p−1) f p p ∗ = ζ = ζ R (p−2) = ζ Möb ψ (p−2) = ψ (p−2) , (c ,(c ) ) ∗ µ ∗ ∗ µ µ ∞ (p−1) n i.e., f(cp,(cp)∗)(z) = Cn z . nP=1 In addition the computations show that

∞ ∗ p (p) n (4.1) ζ (z) = Cn z nX=1 for every p > 2. Using the combinatorial Fourier Transform invented in [9] we can also compute Möb∗ p for any natural number p: let denote the Fourier F transform on formal power series without constant terms and with non-vanishing first coefficients defined as follows (cf. [9]):

z f(z) = f −1(z) F (here f −1 denotes the inverse with respect to composition.) Then (f g) = f F ∗ F · g for all power series f and g in the domain of . Also ζ(z) = 1/(1 + z) hence F F F (ζ∗ p)(z) = 1/(1 + z)p and (Möb∗ p)(z) = (1 + z)p. Thus Möb∗ p(z) = (z(1 + F F Powers of R-diagonal elements 211 z)p)h−1i, and Lagranges Inversion Formula applies to compute the coefficients in ∞ ∗ p ∗ p n Möb : Let Möb (z) = αnz , then nP=1 1 1 n 1 ∞ pn α = Res , z = 0 = Res z−n − zj, z = 0 n n z(1 + z)p n j Xj=1 1 pn = − =: C(−p) n n 1 n − and thus

∞ ∗ p (−p) n (4.2) M¨ob (z) = Cn z . nX=1

Following formulas (4.1) and (4.2), we have Möb = ζ ∗ (−1) and Möb∗ p = ζ∗ (−p) for all natural numbers p. The coefficients in the series ζ∗ p was also computed in [13].

Acknowledgements. The author would like to thank F. Lehner for reading pre- liminary versions of this paper. Also the author would like to thank U. Haagerup for suggesting the problems treated in this paper and for many fruitful conversations.

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FLEMMING LARSEN IMADA University of South Denmark Odense University Campus Campusvej 55 DK–5230 Odense M DENMARK E-mail: [email protected]

Received July 28, 1999.