Abstract:

We characterize the similarity of $n$-shift plus certain weighted Volterra operator denoted by $T_{1}$ on the Dirichlet space ${\cal D}$, and prove that $T_{1}$ acting on ${\cal D}$ is similar to the multiplication operator $M_{p}$ acting on $S({\Bbb D})$ by using some techniques in operator theory. Furthermore, we consider the case of $p(z)=z^{n}$ corresponding to the operator $T_{2}$, and prove that the operator $T_{2}$ has exactly $2^{n}$ reducing subspaces.

Key words: Dirichlet space, Volterra operator, n-Shift operator, Reducing subspaces