An introduction to Rota’s universal operators: properties, old and new examples and future issues

Carl C. Cowen; Eva A. Gallardo-Gutiérrez

Concrete Operators (2016)

  • Volume: 3, Issue: 1, page 43-51
  • ISSN: 2299-3282

Abstract

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The Invariant Subspace Problem for Hilbert spaces is a long-standing question and the use of universal operators in the sense of Rota has been an important tool for studying such important problem. In this survey, we focus on Rota’s universal operators, pointing out their main properties and exhibiting some old and recent examples.

How to cite

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Carl C. Cowen, and Eva A. Gallardo-Gutiérrez. "An introduction to Rota’s universal operators: properties, old and new examples and future issues." Concrete Operators 3.1 (2016): 43-51. <http://eudml.org/doc/277083>.

@article{CarlC2016,
abstract = {The Invariant Subspace Problem for Hilbert spaces is a long-standing question and the use of universal operators in the sense of Rota has been an important tool for studying such important problem. In this survey, we focus on Rota’s universal operators, pointing out their main properties and exhibiting some old and recent examples.},
author = {Carl C. Cowen, Eva A. Gallardo-Gutiérrez},
journal = {Concrete Operators},
keywords = {Rota’s universal operators; Invariant subspace; Analytic Toeplitz operator; Lomonosov’s Theorem; Rota's universal operators; invariant subspace; analytic Toeplitz operator; Lomonosov's theorem},
language = {eng},
number = {1},
pages = {43-51},
title = {An introduction to Rota’s universal operators: properties, old and new examples and future issues},
url = {http://eudml.org/doc/277083},
volume = {3},
year = {2016},
}

TY - JOUR
AU - Carl C. Cowen
AU - Eva A. Gallardo-Gutiérrez
TI - An introduction to Rota’s universal operators: properties, old and new examples and future issues
JO - Concrete Operators
PY - 2016
VL - 3
IS - 1
SP - 43
EP - 51
AB - The Invariant Subspace Problem for Hilbert spaces is a long-standing question and the use of universal operators in the sense of Rota has been an important tool for studying such important problem. In this survey, we focus on Rota’s universal operators, pointing out their main properties and exhibiting some old and recent examples.
LA - eng
KW - Rota’s universal operators; Invariant subspace; Analytic Toeplitz operator; Lomonosov’s Theorem; Rota's universal operators; invariant subspace; analytic Toeplitz operator; Lomonosov's theorem
UR - http://eudml.org/doc/277083
ER -

References

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  1. [1] A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math. 81(1949), 239–255.  Zbl0033.37701
  2. [2] S. R. Caradus, Universal operators and invariant subspaces, Proc. Amer. Math. Soc. 23(1969), 526–527.  Zbl0186.19204
  3. [3] I. Chalendar and J. R. Partington, Modern Approaches to the Invariant Subspace Problem, Cambridge University Press, 2011.  Zbl1231.47005
  4. [4] C. C. Cowen, The commutant of an analytic Toeplitz operator, Trans. Amer. Math. Soc. 239(1978), 1–31.  Zbl0391.47014
  5. [5] C. C. Cowen, The commutant of an analytic Toeplitz operator, II, Indiana Math. J. 29(1980), 1–12.  Zbl0408.47024
  6. [6] C. C. Cowen, An analytic Toeplitz operator that commutes with a compact operator and a related class of Toeplitz operators, J. Functional Analysis 36(1980), 169–184.  Zbl0438.47029
  7. [7] C. C. Cowen and E. A. Gallardo-Gutiérrez, Unitary equivalence of one-parameter groups of Toeplitz and composition operators, J. Functional Analysis 261(2011), 2641–2655.  Zbl1242.47022
  8. [8] C. C. Cowen and E. A. Gallardo-Gutiérrez, Rota’s universal operators and invariant subspaces in Hilbert spaces, to appear.  
  9. [9] C. C. Cowen and E. A. Gallardo-Gutiérrez, Consequences of Universality Among Toeplitz Operators, J. Math. Anal. Appl. 432(2015), 484–503.  
  10. [10] R. G. Douglas, Banach Algebra Techniques in Operator Theory, Academic Press, New York, 1972.  Zbl0247.47001
  11. [11] R. G. Douglas, H. S. Shapiro, and A. L. Shields, Cyclic vectors and invariant subspaces for the backward shift operator, Ann. Inst. Fourier (Grenoble) 20(1970), 37–76.  Zbl0186.45302
  12. [12] P. L. Duren Theory of Hp Spaces, Academic Press, New York, 1970; reprinted with supplement by Dover Publications, Mineola, N˙Y ˙ , 2000.  
  13. [13] Enflo, P., On the invariant subspace problem in Banach spaces, Acta Math. 158(1987), 213–313.  Zbl0663.47003
  14. [14] E. A. Gallardo-Gutiérrez and P. Gorkin, Minimal invariant subspaces for composition operators, J. Math. Pure Appl. 95(2011), 245–259.  Zbl1213.47007
  15. [15] D. W. Hadwin, E. A. Nordgren, H. Radjavi and P. Rosenthal, An operator not satisfying Lomonosov hypotheses, J. Functional Analysis 38(1980), 410–415.  Zbl0451.47003
  16. [16] K. Hoffman, Banach spaces of analytic functions, Dover Publication, Inc., 1988.  Zbl0734.46033
  17. [17] V. Lomonosov, On invariant subspaces of families of operators commuting with a completely continuous operator, Funkcional Anal. i Prilozen 7(1973) 55-56 (Russian).  
  18. [18] B. Sz-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space, North-Holland Publishing Co., 1970.  Zbl0201.45003
  19. [19] N. K. Nikolski, Operators, Functions, and Systems: An Easy Reading, Volume 1: Hardy, Hankel and Toeplitz, Mathematical Surveys and Monographs 92, American Mathematical Society, 2002.  
  20. [20] N. K. Nikolski, Personal communication.  
  21. [21] E. A. Nordgren, P. Rosenthal, and F. S. Wintrobe, Composition operators and the invariant subspace problem, C. R. Mat. Rep. Acad. Sci. Canada, 6(1984), 279–282.  Zbl0599.47041
  22. [22] E. A. Nordgren, P. Rosenthal, and F. S. Wintrobe, Invertible composition operators on Hp, J. Functional Analysis 73(1987), 324– 344.  Zbl0643.47034
  23. [23] J. R. Partington and E. Pozzi, Universal shifts and composition operators, Oper. Matrices 5(2015), 455–467.  Zbl1244.47007
  24. [24] H. Radjavi and P. Rosenthal, Invariant subspaces, Springer-Verlag, New York, 1973.  
  25. [25] Read, C. J., A solution to the invariant subspace problem on the space `1, Bull. London Math. Soc. 17(1985), 305–317.  Zbl0574.47006
  26. [26] Read, C. J., The invariant subspace problem for a class of Banach spaces. II. Hypercyclic operators, Israel J. Math. 63(1988), 1–40.  Zbl0782.47002
  27. [27] G.-C. Rota, On models for linear operators, Comm. Pure Appl. Math. 13(1960), 469–472.  Zbl0097.31604

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