Integral Equations and Operator Theory

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

The 3-Isometric Lifting Theorem

  • Scott McCullough
  • Benjamin Russo


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) 


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


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agler J.: The Arveson extension theorem and coanalytic models. Integral Equ. Oper. Theory 5(5), 608–631 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Agler, J.: Subjordan operators. PhD Thesis, Indiana University (1980)Google Scholar
  3. 3.
    Agler J., Stankus M.: m-isometric transformations of Hilbert space II. Integral Equ. Oper. Theory 23(1), 1–48 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Agler J., Stankus M.: m-isometric transformations of Hilbert space I. Integral Equ. Oper. Theory 21(4), 383–429 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Agler J., Stankus M.: m-isometric transformations of Hilbert space III. Integral Equ. Oper. Theory 24(4), 379–421 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ball J.A., Helton J.W.: Nonnormal dilations, disconjugacy and constrained spectral factorization. Integral Equ. Oper. Theory 3(2), 216–309 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 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)Google Scholar
  8. 8.
    Bermdez T., Martinn A., Negrn E.: Weighted shift operators which are m-isometries. Integral Equ. Oper. Theory 68(3), 301–312 (2010)CrossRefGoogle Scholar
  9. 9.
    Bermdez, T., Martinn, A., Mller, V., Noda, J.A.: Perturbation of m-isometries by nilpotent operators. Abstr. Appl. Anal. Art. ID 745479 (2014)Google Scholar
  10. 10.
    Gleason J., Richter S.: m-isometric commuting tuples of operators on a Hilbert space. Integral Equ. Oper. Theory 56(2), 181–196 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Helton J.W.: Jordan operators in infinite dimensions and Sturm–Liouville conjugate point theory. Bull. Am. Math. Soc. 78, 57–61 (1971)MathSciNetCrossRefGoogle Scholar
  13. 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)Google Scholar
  14. 14.
    Helton J.W.: Infinite dimensional Jordan operators and Sturm–Liouville conjugate point theory. Trans. Am. Math. Soc. 170, 305–331 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    McCullough S.: Sub-Brownian operators. J. Oper. Theory 22(2), 291–305 (1989)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Patton L.J., Robbins M.E.: Composition operators that are m-isometries. Houston J. Math. 31(1), 255–266 (2005)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Paulsen V.: Completely Bounded Maps and Operator Algebras, Cambridge Studies in Advanced Mathematics, vol. 78. Cambridge University Press, Cambridge (2002)Google Scholar
  18. 18.
    Richter S.: A representation theorem for cyclic analytic two-isometries. Trans. Am. Math. Soc. 328(1), 325–349 (1991)CrossRefzbMATHGoogle Scholar
  19. 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. 20.
    Stankus M.: m-isometries, n-symmetries and other linear transformations which are hereditary roots. Integral Equ. Oper. Theory 75(3), 301–321 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of FloridaGainesvilleUSA

Personalised recommendations