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2016 | 3 | 1 | 43-51
Tytuł artykułu

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

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Czasopismo
Rocznik
Tom
3
Numer
1
Strony
43-51
Opis fizyczny
Daty
otrzymano
2015-12-03
zaakceptowano
2016-01-26
online
2016-04-14
Twórcy
  • Indiana University-Purdue University Indianapolis, USA
  • Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid e ICMAT, Plaza de Ciencias 3, 28040, Madrid, Spain
Bibliografia
  • [1] A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math. 81(1949), 239–255.
  • [2] S. R. Caradus, Universal operators and invariant subspaces, Proc. Amer. Math. Soc. 23(1969), 526–527.
  • [3] I. Chalendar and J. R. Partington, Modern Approaches to the Invariant Subspace Problem, Cambridge University Press, 2011.
  • [4] C. C. Cowen, The commutant of an analytic Toeplitz operator, Trans. Amer. Math. Soc. 239(1978), 1–31.
  • [5] C. C. Cowen, The commutant of an analytic Toeplitz operator, II, Indiana Math. J. 29(1980), 1–12.
  • [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.
  • [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.
  • [8] C. C. Cowen and E. A. Gallardo-Gutiérrez, Rota’s universal operators and invariant subspaces in Hilbert spaces, to appear.
  • [9] C. C. Cowen and E. A. Gallardo-Gutiérrez, Consequences of Universality Among Toeplitz Operators, J. Math. Anal. Appl. 432(2015), 484–503.
  • [10] R. G. Douglas, Banach Algebra Techniques in Operator Theory, Academic Press, New York, 1972.
  • [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.
  • [12] P. L. Duren Theory of Hp Spaces, Academic Press, New York, 1970; reprinted with supplement by Dover Publications, Mineola, N˙Y ˙ , 2000.
  • [13] Enflo, P., On the invariant subspace problem in Banach spaces, Acta Math. 158(1987), 213–313.
  • [14] E. A. Gallardo-Gutiérrez and P. Gorkin, Minimal invariant subspaces for composition operators, J. Math. Pure Appl. 95(2011), 245–259.
  • [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.
  • [16] K. Hoffman, Banach spaces of analytic functions, Dover Publication, Inc., 1988.
  • [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] B. Sz-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space, North-Holland Publishing Co., 1970.
  • [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] N. K. Nikolski, Personal communication.
  • [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.
  • [22] E. A. Nordgren, P. Rosenthal, and F. S. Wintrobe, Invertible composition operators on Hp, J. Functional Analysis 73(1987), 324– 344.
  • [23] J. R. Partington and E. Pozzi, Universal shifts and composition operators, Oper. Matrices 5(2015), 455–467.
  • [24] H. Radjavi and P. Rosenthal, Invariant subspaces, Springer-Verlag, New York, 1973.
  • [25] Read, C. J., A solution to the invariant subspace problem on the space `1, Bull. London Math. Soc. 17(1985), 305–317.
  • [26] Read, C. J., The invariant subspace problem for a class of Banach spaces. II. Hypercyclic operators, Israel J. Math. 63(1988), 1–40.
  • [27] G.-C. Rota, On models for linear operators, Comm. Pure Appl. Math. 13(1960), 469–472.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_1515_conop-2016-0006
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