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Integral Equations and Operator Theory

, Volume 84, Issue 1, pp 69–87 | Cite as

The 3-Isometric Lifting Theorem

  • Scott McCullough
  • Benjamin Russo
Article
  • 130 Downloads

Abstract

An operator T on Hilbert space is a 3-isometry if \({T^{*n}T^{n}= I +n B_1 +n^{2} B_2}\) is quadratic in n. An operator J is a Jordan operator if J = U + N where U is unitary, N 2 = 0 and U and N commute. If T is a 3-isometry and \({c > 0,}\) then \({I-c^{-2} B_{2} + sB_{1} + s^{2}B_2}\) 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.

Mathematics Subject Classification

47A20 (Primary) 47A45 47B99 34B24 (Secondary) 

Keywords

Dilation theory 3-symmetric operators 3-isometric operators non-normal spectral theory complete positivity Wiener–Hopf factorization 

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Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of FloridaGainesvilleUSA

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