The Isometry, Compactness And Similarity Of Certain Multiplication Operator On Sobolev Disk Algebra

LetΩ(?)C be a bounded simply connected analytic Cauchy domain, dA denote the planar Lebesgue measure and W2,2(Ω)={f∈L2(Ω,dA):the distributional partial derivatives of first and second order of f belong to L2(Ω,dA)} be the Sobolev space. Let the inner product on W2,2(Ω) be and R(Q) be the closure of all rational functions with poles outsideΩin W2,2(Ω). IfΩ=D={z∈C:|z|<1},R(D) is said to be the Sobolev disk algebra.In this paper, we discuss the isometry, compactness and similarity of certain multiplication operator on sobolev disk algebra, give some necessary and sufficient conditions about that certain multiplication operator Mφis the isometric operator and compact operator, and show that if f,g∈R(Q) and f is a conformal mapping, then multiplication operator Mf and Mg acting on R(Ω) are similar if and only if there exist a conformal mappingφ:Ω→D, z0∈D,|c|=1 such that whereφz0(z)=c(z-z0)/(1-z0z). In addition, under some restrict conditions, we also give a necessary and sufficient condition about the similarity of the multiplication operator Mf, Mg on R(D), and a sufficient condition about that multiplication operator Mz is similar to Mz+K, where K is a bounded operator on R(D) with strictly lower-triangular operator matrix representation.