MULTIPLICATION OPERATORS ON WEIGHTED DIRICHLET SPACES

In this paper,we study multiplication operators on weighted Dirichlet spaces Dβ（β∈R）.Let n be a positive integer and β∈R,we show that the multiplication operator Mz~n on Dβ is similar to the operator ⊕1~nMz on the space ⊕1~nDβ.Moreover,we prove that Mz~n（≥2） on Dβ is unitarily equivalent to ⊕1~nMz on ⊕1~nDβ if and only if β=0.In addition,we completely characterize the unitary equivalence of the restrictions of Mz~n to different invariant subspaces zkDβ（k≥1）,and the unitary equivalence of the restrictions of Mz~n to different invariant subspaces Sj（0≤jz~n) of Mz~n on a family of analytic function spaces Aα2（α ∈ R） on D(in fact,the family of spaces Aα2（α∈R） is the same with the family of spaces Dβ（β∈R）) in terms of the multiplier algebra of the underlying function spaces.In this paper,we give a new characterization of the commutant A'(Mz~n) of Mz~n on Dβ,and characterize the self-adjoint operators and unitary operators in A'(Mz~n).We find that the class of self-adjoint operators（unitary operators） in A'(Mz~n) when β≠0 is different from the class of self-adjoint operators（unitary operators） in A'(Mz~n) when β=0.