Berger Measure for Some Subnormal Shifts
A subnormal weighted shift may be transformed to another shift in various ways, such as taking the p-th power of each weight or forming the Aluthge transform. We determine in a number of cases whether the resulting shift is subnormal, and, if it is, find a concrete representation of the associated Berger measure, directly for finitely atomic measures, and using both Laplace transform and Fourier transform methods for more complicated measures. Alternatively, the problem may be viewed in purely measure-theoretic terms as the attempt to solve moment matching equations such as (int t^n d mu(t))^2 = t^n d nu(t) (n = 0, 1, . . .) for one measure given the other.
Integral Equations Operator Theory
Exner, George R. and Curto, Raul. "Berger Measure for Some Subnormal Shifts." Integral Equations Operator Theory (2016) : 429-450.