The 3-Isometric Lifting Theorem

An operator T on Hilbert space is a 3-isometry if T∗nTn=I+nB1+n2B2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T^{*n}T^{n}= I +n B_1 +n^{2} B_2}$$\end{document} is quadratic in n. An operator J is a Jordan operator if J = U + N where U is unitary, N2 = 0 and U and N commute. If T is a 3-isometry and c>0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c > 0,}$$\end{document} then I-c-2B2+sB1+s2B2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${I-c^{-2} B_{2} + sB_{1} + s^{2}B_2}$$\end{document} is positive semidefinite for all real s if and only if it is the restriction of a Jordan operator J = U + N with the norm of N at most c. As a corollary, an analogous result for 3-symmetric operators, due to Helton and Agler, is recovered.