The 3-Isometric Lifting Theorem

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.

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Correspondence to Scott McCullough.

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S. McCullough was partially supported by NSF Grants DMS-1101137 and DMS-1361501.

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McCullough, S., Russo, B. The 3-Isometric Lifting Theorem. Integr. Equ. Oper. Theory 84, 69–87 (2016). https://doi.org/10.1007/s00020-015-2240-7

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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