I have seen that Toeplitz operators are defined only on weighted Bergman spaces. Can someone tell me how can we (if we can) define them on weighted Dirichlet spaces?
Recall that for $\beta\in\mathbb{R}$, the weighted Dirichlet space $\mathcal{D}_\beta$ is : $$\mathcal{D}_\beta:=\{f\in \mathcal{H}(\mathbb{D}), f(z)=\sum_{n=0}^{\infty}a_nz^n: \quad\rVert f\lVert_\beta^2:=\sum_{n=0}^{\infty}|a_n|^2(n+1)^{2\beta}<\infty \}.$$
Note that weighted Bergman spaces coincide with weighted Dirichlet spaces whenever $\beta<0$.
My question is the following:
How can we define Toeplitz operator on $\mathcal{D}_\beta$ for $\beta>0$?
Thank you.