We will research the special operator and operator algebras on Hardy-Sobolev spaces and Bergman-Sobolev spaces, these spaces first occurs in Harmonic analysis. Recently, the complex analytic construction and  the Sobolev space construction are connected, since the complex analytic function spaces have better construction than real function spaces, the properties and construction of operators and operator algebras on these spaces are deeper than that in the case of real variables. However, the construction of these spaces are more complex than that of classical Hardy and Bergman spaces, in fact, the classical methods and techniques almost fail on these spaces,  the theory of operators and operator algebras on the spaces seems very difficult. We will continue the research of properties of the multipliers, Toeplitz operators, composition operators and constructions of generated algebras on the spaces