Weighted Composition-differentiation operators on weighted Dirichlet spaces

Abstract

We study weighted composition-differentiation operators of order $n,$ $D_{\psi ,\phi }^{n}f:=\psi \cdot f^{(n)}\circ \phi$ on weighted Dirichlet spaces. Our main goal is to describe the operator-theoretic properties of $D_{\psi ,\phi }^n$ in terms of the inducing functions $\psi$ and $\phi .$ We characterize bounded, compact, and Hilbert-Schmidt class of such operators and estimate their essential norms. Our approach relies on Carleson measures and the asymptotic behavior of inducing functions. Furthermore, we prove that $D_{\psi ,\phi }^n$ is bounded for all $\psi$ in the space precisely when $\Vert \phi {\Vert }_{\infty }<1$ and $\phi$ belongs to the multiplier algebra of the space. Finally, we give norm estimates for particular elliptic inducing functions and provide examples to demonstrate our results.