Generalized Markov–Stieltjes operator on Hardy and Lebesgue spaces

Generalized Markov–Stieltjes operator $S_\alpha$ is introduced, inverse formulas and operational properties for corresponding integral transform obtained, boundedness, nuclearity or compactness of operator $S_\alpha$ on Hardy and Lebesgue spaces for various values of complex parameter $\alpha$ proved, its trace is calculated, and estimates for its norm are given.