Invertible weighted composition operators preserve frames on Dirichlet type spaces

Abstract

Some characterizations for weighted composition operators to be invertible on Dirichlet type spaces $\mathfrak{D}_{\rho}$ are given in this paper when $\rho$ is finite lower type greater than $0$ and upper type less than $1$. In particular, the equivalence between invertible and preserve frames is established. Moreover, weighted composition operators that preserve tight frames and normalized tight frames on the Dirichlet type space $\mathfrak{D}_{\alpha}$ $(0 < \alpha < 1)$ are also investigated.

Keywords

• dirichlet type space
• weighted composition operator
• frame
• bounded below
• multiplier