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Integral Equations and Operator Theory

, Volume 75, Issue 3, pp 301–321 | Cite as

m-Isometries, n-Symmetries and Other Linear Transformations Which are Hereditary Roots

  • Mark StankusEmail author
Article

Abstract

We develop techniques to study bounded linear transformations T such that \({\sum_{m,n} c_{m,n} T^{*n}T^m = 0}\) where \({c_{m,n} \in {\bf C}}\) and all but finitely many c m,n are zero. Such an operator is called a hereditary root of the polynomial \({p(x,y) = \sum_{m,n} c_{m,n} y^{n}x^{m}}\) . Self-adjoint operators, isometries, n-symmetric operators and m-isometries are all examples of roots of polynomials. The techniques developed include finding maximal invariant subspaces satisfying certain operator-theoretic properties, descriptions of the spectral picture of a root, resolvent inequalities for a root, and additional properties which can be derived when knowing more about the spectrum of a particular root. The results involving maximal invariant subspaces are applicable to operators which are not roots of polynomials.

Mathematics Subject Classification (2010)

Primary 47A60 Secondary 47A99 

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References

  1. 1.
    Agler, J.: Subjordan operators, Thesis. Indiana University (1980)Google Scholar
  2. 2.
    Agler, J.: An abstract approach to model theory. Survey of some recent results in operator theory, vol. II. Pitman Research Notes in Mathematics Series, No. 192Google Scholar
  3. 3.
    Agler J.: A disconjugacy theorem for Toeplitz operators. Am. J. Math. 112, 1–14 (1980)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Agler J.: Hypercontractions and subnormality. J. Oper. Theory 13, 203–217 (1985)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Agler J.: The Arveson extension theorem and coanalytic models. J. Integr. Equ. Oper. Theory 5, 608–631 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Agler J.: Subjordan operators: Bishop’s theorem, spectral inclusion, and spectral sets. J. Oper. Theory 7, 373–395 (1982)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Agler J.: Rational dilation of an annulus. Ann. Math. 121, 537–563 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Agler J., Helton W.J., Stankus M.: Classification of hereditary matrices. Linear Algebra Appl. 274, 125–160 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Agler J., Stankus M.: m-isometric transformations on Hilbert Space I. J. Integr. Equ. Oper. Theory 21(4), 383–429 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Agler J., Stankus M.: m-isometric transformations on Hilbert space II. J. Integr. Equ. Oper. Theory 23(1), 1–48 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Agler J., Stankus M.: m-isometric linear transformations on Hilbert space III. J. Integr. Equ. Oper. Theory 24(4), 379–421 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Berberian S.K.: Approximate proper values. Proc. Am. Math. Soc. 13, 111–114 (1962)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Conway J.B.: A Course in Functional Analysis. Springer, New York (1985)zbMATHGoogle Scholar
  14. 14.
    Curto R.E., Putinar M.: Existence of non-subnormal polynomially hyponormal operators. B.A.M.S. 25(2), 373–378 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Helton, J.W.: Operators with a representation as multiplication by x on a Sobolev space. Colloquia Math. Soc. Janos Bolyai 5, Hilbert Space Operators, Tihany, pp. 279–287 (1970)Google Scholar
  16. 16.
    McCullough, S.A.: 3-Isometries, Thesis. University of California, San Diego (1987)Google Scholar
  17. 17.
    McCullough S.A.: Sub Brownian operators. J. Oper. Theory 22(2), 291–305 (1989)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Richter S.: Invariant subspaces of the Dirichlet shift. J. Reine Agnew. Math. 386, 205–220 (1988)zbMATHGoogle Scholar
  19. 19.
    Radjavi H., Rosenthal P.: On roots of normal operators. J. Math. Anal. Appl. 34, 653–664 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Radjavi H., Rosenthal P.: Invariant Subspaces. Springer, New York (1973)zbMATHCrossRefGoogle Scholar
  21. 21.
    Stankus, M.: Isosymmetric Linear Transformations on Complex Hilbert Space, Thesis (Ph.D.). University of California, San Diego, pp. 1–80 (1993)Google Scholar

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© Springer Basel 2013

Authors and Affiliations

  1. 1.California Polytechnic State UniversitySan Luis ObispoUSA

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