Fredholm Weighted Generalized Composition Operators

In the present paper, properties of the self map on D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {D}$$\end{document}, the open unit disc, and complex valued mapping on D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {D}$$\end{document} are obtained when the induced kth-order weighted generalized composition operator on weighted Hardy space is Fredholm. We also describe the action of the adjoint of generalized multiplication, generalized composition and kth\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k{th}$$\end{document}-order weighted generalized composition operators on the derivative evaluation kernel functions, which in turn provide some invariant subspaces for these operators.