A Classification of \({\varvec{n}}\)-Tuples of Commuting Shifts of Finite Multiplicity

  • Edward J. Timko


Let \(\mathbb {V}\) denote an n-tuple of shifts of finite multiplicity, and denote by \({{\mathrm{Ann}}}(\mathbb {V})\) the ideal consisting of polynomials p in n complex variables such that \(p(\mathbb {V})=0\). If \(\mathbb {W}\) on \(\mathfrak {K}\) is another n-tuple of shifts of finite multiplicity, and there is a \(\mathbb {W}\)-invariant subspace \(\mathfrak {K}'\) of finite codimension in \(\mathfrak {K}\) so that \(\mathbb {W}|\mathfrak {K}'\) is similar to \(\mathbb {V}\), then we write \(\mathbb {V}\lesssim \mathbb {W}\). If \(\mathbb {W}\lesssim \mathbb {V}\) as well, then we write \(\mathbb {W}\approx \mathbb {V}\). In the case that \({{\mathrm{Ann}}}(\mathbb {V})\) is a prime ideal we show that the equivalence class of \(\mathbb {V}\) is determined by \({{\mathrm{Ann}}}(\mathbb {V})\) and a positive integer k. More generally, the equivalence class of \(\mathbb {V}\) is determined by \({{\mathrm{Ann}}}(\mathbb {V})\) and an m-tuple of positive integers, where m is the number of irreducible components of the zero set of \({{\mathrm{Ann}}}(\mathbb {V})\).


Constrained operators Commuting pure isometries Virtual similarity 

Mathematics Subject Classification



  1. 1.
    Abrahamse, M.B., Douglas, R.G.: A class of subnormal operators related to multiply-connected domains. Adv. Math. 19(1), 106–148 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Agler, J., Knese, G., McCarthy, J.E.: Algebraic pairs of isometries. J. Oper. Theory 67(1), 215–236 (2012)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Agler, J., McCarthy, J.E.: Distinguished varieties. Acta Math. 194, 133–153 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ahern, P.R., Sarason, D.: On some hypo-Dirichlet algebras of analytic functions. Am. J. Math. 89, 932–941 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bungart, L.: On analytic fiber bundles. I. Holomorphic fiber bundles with infinite dimensional fibers. Topology 7, 55–68 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Farkas, H., Kra, I.: Riemann surfaces. In: Axler, S., Gehring, F.W., Ribet, K.A. (eds.) Graduate Texts in Mathematics, vol. 71, 2nd edn. Springer, New York (1992)Google Scholar
  7. 7.
    Gamelin, T.: Embedding Riemann surfaces in maximal ideal spaces. J. Funct. Anal. 2, 123–146 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gamelin, T., Lumer, G.: Theory of abstract Hardy spaces and the universal Hardy class. Adv. Math. 2, 118–174 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gamelin, T.: Uniform Algebras. Prentice-Hall Inc, Englewood Cliffs (1969)zbMATHGoogle Scholar
  10. 10.
    Gunning, R.: Lectures on Riemann Surfaces. Princeton Mathematical Notes. Princeton University Press, Princeton (1966)zbMATHGoogle Scholar
  11. 11.
    Helson, H.: Lectures on Invariant Subspaces. Academic Press, New York (1964)zbMATHGoogle Scholar
  12. 12.
    Kendig, K.: Elementary Algebraic Geometry. Graduate Texts in Mathematics, vol. 44. Springer, New York (1977)CrossRefzbMATHGoogle Scholar
  13. 13.
    Khavinson, S.Y.: Theory of factorization of single-valued analytic functions on compact Riemann surfaces with a boundary. Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk 44 (1989) no. 4(268) 155–189, 256; translation. Russian Math. Surveys 44(4), 113–156 (1989)Google Scholar
  14. 14.
    Nagy, B.S., Foias, C., Bercovici, H., Krchy, L.: Harmonic Analysis of Operators on Hilbert Space. Revised and Enlarged Edition. Universitext, 2nd edn. Springer, New York (2010)CrossRefGoogle Scholar
  15. 15.
    Rudin, W.: Analytic functions of class \(H^p\). Trans. Am. Math. Soc. 78, 46–66 (1955)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Stout, E.L.: On some algebras of analytic functions on finite open Riemann surfaces. Math. Z. 92, 366–379 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Timko, E.: On polynomial \(n\)-tuples of commuting isometries. J. Oper. Theory 77, 391–420 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Tsuji, M.: Potential Theory in Modern Function Theory. Reprinting of the 1959, p. 1975. Chelsea Publishing Co., New York (1959)Google Scholar
  19. 19.
    Wermer, J.: Analytic disks in maximal ideal spaces. Am. J. Math. 86, 161–170 (1964)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Wijesooriya, U.: Finite Rank Isopairs. arxiv:1610.02602

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Authors and Affiliations

  1. 1.University of ManitobaWinnipegCanada

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