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.
References
[1] A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math. 81(1949), 239–255. 10.1007/BF02395019Search in Google Scholar
[2] S. R. Caradus, Universal operators and invariant subspaces, Proc. Amer. Math. Soc. 23(1969), 526–527. 10.1090/S0002-9939-1969-0250104-4Search in Google Scholar
[3] I. Chalendar and J. R. Partington, Modern Approaches to the Invariant Subspace Problem, Cambridge University Press, 2011. 10.1017/CBO9780511862434Search in Google Scholar
[4] C. C. Cowen, The commutant of an analytic Toeplitz operator, Trans. Amer. Math. Soc. 239(1978), 1–31. 10.1090/S0002-9947-1978-0482347-9Search in Google Scholar
[5] C. C. Cowen, The commutant of an analytic Toeplitz operator, II, Indiana Math. J. 29(1980), 1–12. 10.1512/iumj.1980.29.29001Search in Google Scholar
[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. 10.1016/0022-1236(80)90098-1Search in Google Scholar
[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. 10.1016/j.jfa.2011.07.005Search in Google Scholar
[8] C. C. Cowen and E. A. Gallardo-Gutiérrez, Rota’s universal operators and invariant subspaces in Hilbert spaces, to appear. Search in Google Scholar
[9] C. C. Cowen and E. A. Gallardo-Gutiérrez, Consequences of Universality Among Toeplitz Operators, J. Math. Anal. Appl. 432(2015), 484–503. 10.1016/j.jmaa.2015.06.061Search in Google Scholar
[10] R. G. Douglas, Banach Algebra Techniques in Operator Theory, Academic Press, New York, 1972. Search in Google Scholar
[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. 10.5802/aif.338Search in Google Scholar
[12] P. L. Duren Theory of Hp Spaces, Academic Press, New York, 1970; reprinted with supplement by Dover Publications, Mineola, N˙Y ˙ , 2000. Search in Google Scholar
[13] Enflo, P., On the invariant subspace problem in Banach spaces, Acta Math. 158(1987), 213–313. 10.1007/BF02392260Search in Google Scholar
[14] E. A. Gallardo-Gutiérrez and P. Gorkin, Minimal invariant subspaces for composition operators, J. Math. Pure Appl. 95(2011), 245–259. 10.1016/j.matpur.2010.04.003Search in Google Scholar
[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. 10.1016/0022-1236(80)90073-7Search in Google Scholar
[16] K. Hoffman, Banach spaces of analytic functions, Dover Publication, Inc., 1988. Search in Google Scholar
[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). 10.1007/BF01080698Search in Google Scholar
[18] B. Sz-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space, North-Holland Publishing Co., 1970. Search in Google Scholar
[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. Search in Google Scholar
[20] N. K. Nikolski, Personal communication. Search in Google Scholar
[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. Search in Google Scholar
[22] E. A. Nordgren, P. Rosenthal, and F. S. Wintrobe, Invertible composition operators on Hp, J. Functional Analysis 73(1987), 324– 344. 10.1016/0022-1236(87)90071-1Search in Google Scholar
[23] J. R. Partington and E. Pozzi, Universal shifts and composition operators, Oper. Matrices 5(2015), 455–467. Search in Google Scholar
[24] H. Radjavi and P. Rosenthal, Invariant subspaces, Springer-Verlag, New York, 1973. 10.1007/978-3-642-65574-6Search in Google Scholar
[25] Read, C. J., A solution to the invariant subspace problem on the space `1, Bull. London Math. Soc. 17(1985), 305–317. 10.1112/blms/17.4.305Search in Google Scholar
[26] Read, C. J., The invariant subspace problem for a class of Banach spaces. II. Hypercyclic operators, Israel J. Math. 63(1988), 1–40. 10.1007/BF02765019Search in Google Scholar
[27] G.-C. Rota, On models for linear operators, Comm. Pure Appl. Math. 13(1960), 469–472. 10.1002/cpa.3160130309Search in Google Scholar
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