Hilbert–Pólya Operators in Krein Spaces

Abstract

We construct some class of selfadjoint operators in the Krein spaces consisting of functions onthe straight line \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$ \\{\\operatorname{Re}s=\\frac{1}{2}\\} $\\end{document}.Each of these operators is a rank-one perturbation of a selfadjoint operatorin the corresponding Hilbert spaceand has eigenvalues complex numbers of the form \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$ \\frac{1}{s(1-s)} $\\end{document},where \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$ s $\\end{document} ranges over the set of nontrivial zeros of the Riemann zeta-function.