SCHATTEN CLASSES OF VOLTERRA OPERATORS ON BERGMAN-TYPE SPACES IN THE UNIT BALL

We devote to studying the condition of a holomorphic function g in the complex unit ball B-n so that the Volterra operator T-g : A(alpha)(2) -> A(alpha)(2) belongs to the Schatten p-class. Assuming n >= 2 and alpha > -3, we get the following conclusions 1. For 0 < p <= n, T-g is an element of S-p (A(alpha)(2)) if and only if g is a constant. 2. For n < p < infinity and p(alpha + 1) + 4n > 0, T-g is an element of S-p (A(alpha)(2)) if and only if integral(Bn) ((1 - vertical bar w vertical bar(2))(n+1+alpha+2t) integral(Bn) vertical bar Rg(z)vertical bar(2) dv(alpha+2)(z)/vertical bar 1 - < z, w >vertical bar(2(n+1+alpha+t)))(p/2) d tau(w) < infinity, where t > max{n/p - n+1+alpha/2, n -1/2} and and d tau(w) = (1 - vertical bar w vertical bar(2))(alpha+n+1)dv(w) is the Mobius invariant measure in B-n. Here dv is the normalized Lebesgue measure on B-n so that v(B-n) = 1 and dv(alpha+2)(z) = c alpha+2(1 - vertical bar z vertical bar(2))(alpha+2) dv(z) with a normalized constant c(alpha+2) so that v(alpha+2)(B-n) = 1.