Conjugations on L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} Space on the Real Line

Influenced by PT operators (parity and time reversal operators) the general conjugations acting in the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{L}}^\textbf{2}$$\end{document} space on the real line with the Lebesgue measure are studied. The behaviour of conjugations with respect to multiplication operators and translations is investigated. We consider conjugations defined as the convolution of the Fourier transform of a given unimodular (even) function and test functions. We characterize all conjugations commuting (intertwining) with all translation operators on the real line. The conjugations leaving kernels of Toeplitz operators (a generalization of model spaces) on H2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{H}}^\textbf{2}$$\end{document} space on the upper half plane invariant are also characterized. Analogous results are obtained for conjugations and kernels of Wiener–Hopf operators.