Dilation Theory and Functional Models for Tetrablock Contractions

摘要

Abstract A classical result of Sz.-Nagy asserts that a Hilbert space contraction operator T can be dilated to a unitary Udocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${{mathcal {U}}}$$end{document}, i.e., Tn=PHUn|Hdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$T^n = P_{{mathcal {H}}}{{mathcal {U}}}^n|{{mathcal {H}}}$$end{document} for all n=0,1,2,…documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$n =0,1,2,ldots $$end{document}. A more general multivariable setting for these ideas is the setup where (i) the unit disk is replaced by a domain Ωdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Omega $$end{document} contained in Cddocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${{mathbb {C}}}^d$$end{document}, (ii) the contraction operator T is replaced by an Ωdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Omega $$end{document}-contraction, i.e., a commutative operator d-tuple T=(T1,…,Td)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${{textbf{T}}}= (T_1, ldots , T_d)$$end{document} on a Hilbert space Hdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${{mathcal {H}}}$$end{document} such that ‖r(T1,…,Td)‖L(H)≤supλ∈Ω|r(λ)|documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Vert r(T_1, ldots , T_d) Vert _{{{mathcal {L}}}({{mathcal {H}}})} le sup _{lambda in Omega } | r(lambda ) |$$end{document} for all rational functions with no singularities in Ω¯documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$overline{Omega }$$end{document} and the unitary operator Udocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${{mathcal {U}}}$$end{document} is replaced by an Ωdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Omega $$end{document}-unitary operator tuple, i.e., a commutative operator d-tuple U=(U1,…,Ud)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} use