Algebraic structure of the $ L_2 $ analytic Fourier–Feynman transform associated with Gaussian paths on Wiener space

Abstract

In this paper we study algebraic structures of the classes of the \begin{document}$ L_2 $\end{document} analytic Fourier–Feynman transforms on Wiener space. To do this we first develop several rotation properties of the generalized Wiener integral associated with Gaussian paths. We then proceed to analyze the \begin{document}$ L_2 $\end{document} analytic Fourier–Feynman transforms associated with Gaussian paths. Our results show that these \begin{document}$ L_2 $\end{document} analytic Fourier–Feynman transforms are actually linear operator isomorphisms from a Hilbert space into itself. We finally investigate the algebraic structures of these classes of the transforms on Wiener space, and show that they indeed are group isomorphic.