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The 3-Isometric Lifting Theorem

Integral Equations and Operator Theory volume 84, pages 69–87 (2016)Cite this article

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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References

  1. Agler J.: The Arveson extension theorem and coanalytic models. Integral Equ. Oper. Theory 5(5), 608–631 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  2. Agler, J.: Subjordan operators. PhD Thesis, Indiana University (1980)

  3. Agler J., Stankus M.: m-isometric transformations of Hilbert space II. Integral Equ. Oper. Theory 23(1), 1–48 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  4. Agler J., Stankus M.: m-isometric transformations of Hilbert space I. Integral Equ. Oper. Theory 21(4), 383–429 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  5. Agler J., Stankus M.: m-isometric transformations of Hilbert space III. Integral Equ. Oper. Theory 24(4), 379–421 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ball J.A., Helton J.W.: Nonnormal dilations, disconjugacy and constrained spectral factorization. Integral Equ. Oper. Theory 3(2), 216–309 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ball, J.A., Fanney, T.R.: Closability of differential operators and real sub-Jordan operators. In: Topics in Operator Theory: Ernst D. Hellinger Memorial Volume, pp. 93–156. Oper. Theory Adv. Appl., vol. 48. Birkhuser, Basel (1990)

  8. Bermdez T., Martinn A., Negrn E.: Weighted shift operators which are m-isometries. Integral Equ. Oper. Theory 68(3), 301–312 (2010)

    Article  Google Scholar 

  9. Bermdez, T., Martinn, A., Mller, V., Noda, J.A.: Perturbation of m-isometries by nilpotent operators. Abstr. Appl. Anal. Art. ID 745479 (2014)

  10. Gleason J., Richter S.: m-isometric commuting tuples of operators on a Hilbert space. Integral Equ. Oper. Theory 56(2), 181–196 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gu C., Stankus M.: Some results on higher order isometries and symmetries: Products and sums with a nilpotent operator. Linear Algebra Appl. 469, 500–509 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Helton J.W.: Jordan operators in infinite dimensions and Sturm–Liouville conjugate point theory. Bull. Am. Math. Soc. 78, 57–61 (1971)

    Article  MathSciNet  Google Scholar 

  13. Helton, J.W.: Operators with a representation as multiplication by on a Sobolev space. Hilbert space operators and operator algebras. In: Proc. Internat. Conf., Tihany, pp. 279–287 (1970)

  14. Helton J.W.: Infinite dimensional Jordan operators and Sturm–Liouville conjugate point theory. Trans. Am. Math. Soc. 170, 305–331 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  15. McCullough S.: Sub-Brownian operators. J. Oper. Theory 22(2), 291–305 (1989)

    MathSciNet  MATH  Google Scholar 

  16. Patton L.J., Robbins M.E.: Composition operators that are m-isometries. Houston J. Math. 31(1), 255–266 (2005)

    MathSciNet  MATH  Google Scholar 

  17. Paulsen V.: Completely Bounded Maps and Operator Algebras, Cambridge Studies in Advanced Mathematics, vol. 78. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  18. Richter S.: A representation theorem for cyclic analytic two-isometries. Trans. Am. Math. Soc. 328(1), 325–349 (1991)

    Article  MATH  Google Scholar 

  19. Rosenblum M., Rovnyak J.: Hardy Classes and Operator Theory. Oxford Mathematical Monographs. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York (1985)

    Google Scholar 

  20. Stankus M.: m-isometries, n-symmetries and other linear transformations which are hereditary roots. Integral Equ. Oper. Theory 75(3), 301–321 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Florida, Gainesville, USA

    Scott McCullough & Benjamin Russo

Authors
  1. Scott McCullough
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  2. Benjamin Russo
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Correspondence to Scott McCullough.

Additional information

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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  • DOI: https://doi.org/10.1007/s00020-015-2240-7

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