Realizations of Slice Hyperholomorphic Generalized Contractive and Positive Functions

We introduce generalized Schur functions and generalized positive functions in the setting of slice hyperholomorphic functions and study their realizations in terms of associated reproducing kernel Pontryagin spaces. To this end, we also prove some results in quaternionic functional analysis like an invariant subspace theorem for contractions in a Pontryagin space. We also consider slice hyperholomorphic functions on the half space H+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{H}_{+}}$$\end{document} of quaternions with positive real parts and we study the Hardy space H2(H+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\mathbb{H}_{+})}$$\end{document} and Blaschke products in this framework.