Polynomially Hyponormal Operators
A survey of the theory of k-hyponormal operators starts with the construction of a polynomially hyponorrnal operator which is not subnormal. This is achieved via a natural dictionary between positive functionals on specific convex cones of polynomials and linear bounded operators acting on a Hilbert space, with a distinguished cyclic vector. The class of unilateral weighted shifts provides an optimal framework for studying k-hyponormality. Non-trivial links with the theory of Toeplitz operators on Hardy space are also exposed in detail. A good selection of intriguing open problems, with precise references to prior works and partial solutions, is offered.
Mathematics Subject Classification (2000)Primary 47B20 Secondary 47B35 47B37 46A55 30E05
KeywordsHyponorrnal operator weighted shift moment problem convex cone Toeplitz operator
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- J. Agier, An abstract approach to model theory, in Surveys of Recent Results in Operator Theory, vol. II (J.B. Conway and B.B. Morrel, eds.), Pitman Res. Notes Math. Ser., vol. 192, Longman Sci. Tech., Harlow, 1988, pp. 1–23.Google Scholar
- I. Colojoara and C. Foias, Theory of Generalized Spectral Operators, Mathematics and its Applications, vol. 9, Gordon and Breach, Science Publishers, New York-London-Paris, 1968.Google Scholar
- J.B. Conway, The Theory of Subnormal Operators, Mathematical Surveys and Monographs, vol. 36, American Mathematical Society, Providence, RI, 1991.Google Scholar
- C. Cowen, Hyponormal and subnormal Toeplitz operators, in Surveys of Some Recent Results in Operator Theory, Vol. I (J.B. Conway and B.B. Morrel, eds.), Pitman Res. Notes in Math., vol. 171, Longman Publ. Co. 1988, pp. 155–167.Google Scholar
- R. Curto and W.Y. Lee, Joint hyponormality of Toeplitz pairs, Memoirs Amer. Math. Soc. 150, no. 712, Amer. Math. Soc, Providence, 2001.Google Scholar
- R. Curto and W.Y. Lee, Subnormality and k-hyponormality of Toeplitz operators: A brief survey and open questions, in Operator Theory and Banach Algebras, The Theta Foundation, Bucharest, 2003; pp. 73–81.Google Scholar
- R. Curto and W.Y. Lee, k-hyponormality of finite rank perturbations of unilateral weighted shifts, Trans. Amer. Math. Soc. 357(2005), 4719-4737.Google Scholar
- P.R. Halmos, A Hilbert Space Problem Book, Second edition, Graduate Texts in Mathematics, vol. 19, Springer-Verlag, New York-Berlin, 1982.Google Scholar
- J.W. Helton and M. Putinar, Positive polynomials in scalar and matrix variables, the spectral theorem, and optimization, in Operator Theory, Structured Matrices, and Dilations, pp. 229–306, Theta Ser. Adv. Math. 7, Theta, Bucharest, 2007.Google Scholar