On posinormality of weighted composition-differentiation operators on $H^2(\mathbb{D})$

TL;DR

This paper investigates the conditions under which weighted composition-differentiation operators on Hardy space are posinormal or coposinormal, providing necessary conditions and an adjoint formula.

Contribution

It identifies specific conditions for posinormality of weighted composition-differentiation operators and derives an adjoint formula to analyze their properties.

Findings

Weighted composition-differentiation operators can be posinormal for certain functions.

The paper establishes necessary conditions for posinormality and coposinormality.

An explicit adjoint formula for these operators is derived.

Abstract

In this article, the posinormality and coposinormality of weighted composition-differentiation operators on Hardy space $H^2(\mathbb{D})$ are investigated. It is observed that while a composition-differentiation operator $D_{\phi,n}$ fails to be posinormal, the weighted composition-differentiation operator $D_{\psi,\phi,n}$ can be posinormal for specific choices of $\psi, \phi$. Some necessary conditions are obtained for posinormality and coposinormality of the operator $D_{\psi,\phi,n}$. Furthermore, the adjoint formula for this operator is derived which also helped us to examine some results regarding posinormality of this operator.