Abstract
In this paper we discuss the p-th power problem for measures which is, for a given measure ν and p>0, to find, if it exists, a measure μ such that ∫tndμ(t)=(∫tndν(t))p. Bergman shifts have some results for this problem and we research the problem for generalizations of Bergman shifts. We investigate the Berger measures for the class of weighted shifts S(a,b,c,d) with a weight sequence {an+bcn+d}n=0∞ (a,b,c,d>0 and ad>bc). Using the square root measure, we discuss also the Berger measures of the Aluthge transform. This problem is related to subnormal weighted shifts and these results add to the extremely small library of subnormal weighted shifts for which the Berger measure is known concretely.
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