Small eigenvalues of large Hermitian moment matrices

We consider an infinite Hermitian positive definite matrix M which is the moment matrix associated with a measure μ with infinite and compact support on the complex plane. We prove that if the polynomials are dense in L 2 ( μ ) then the smallest eigenvalue λ n of the truncated matrix M n of M of size ( n + 1 ) × ( n + 1 ) tends to zero when n tends to infinity. In the case of measures in the closed unit disk we obtain some related results.