Hyponormality of Toeplitz operators
HTML articles powered by AMS MathViewer
- by Carl C. Cowen
- Proc. Amer. Math. Soc. 103 (1988), 809-812
- DOI: https://doi.org/10.1090/S0002-9939-1988-0947663-4
- PDF | Request permission
Abstract:
For $\varphi$ in ${L^\infty }(\partial {\mathbf {D}})$, let $\varphi = f + \bar g$ where $f$ and $g$ are in ${H^2}$. In this note, it is shown that the Toeplitz operator $T\varphi$ is hyponormal if and only if $g = c + {T_{\bar h}}f$ for some constant $c$ and some function $h$ in ${H^\infty }({\mathbf {D}})$ with ${\left \| h \right \|_\infty } \leq 1$.References
- M. B. Abrahamse, Subnormal Toeplitz operators and functions of bounded type, Duke Math. J. 43 (1976), no.Β 3, 597β604. MR 428097
- Ichiro Amemiya, Takashi Ito, and Tin Kin Wong, On quasinormal Toeplitz operators, Proc. Amer. Math. Soc. 50 (1975), 254β258. MR 410451, DOI 10.1090/S0002-9939-1975-0410451-2
- Arlen Brown and P. R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963/64), 89β102. MR 160136, DOI 10.1007/978-1-4613-8208-9_{1}9
- Carl C. Cowen and John J. Long, Some subnormal Toeplitz operators, J. Reine Angew. Math. 351 (1984), 216β220. MR 749683
- Carl C. Cowen, More subnormal Toeplitz operators, J. Reine Angew. Math. 367 (1986), 215β219. MR 839133, DOI 10.1515/crll.1986.367.215 β, Hyponormal and subnormal Toeplitz operators, Surveys of Recent Results in Operator Theory, vol. 1 (J. B. Conway and B. B. Morrel, eds.) (to appear).
- P. R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76 (1970), 887β933. MR 270173, DOI 10.1090/S0002-9904-1970-12502-2
- P. R. Halmos, Ten years in Hilbert space, Integral Equations Operator Theory 2 (1979), no.Β 4, 529β564. MR 555777, DOI 10.1007/BF01691076
- Takashi ItΓ΄ and Tin Kin Wong, Subnormality and quasinormality of Toeplitz operators, Proc. Amer. Math. Soc. 34 (1972), 157β164. MR 303334, DOI 10.1090/S0002-9939-1972-0303334-7 J. J. Long, Hyponormal Toeplitz operators and weighted shifts, Thesis, Michigan State Univ., 1984.
- Donald E. Marshall, Blaschke products generate $H^{\infty }$, Bull. Amer. Math. Soc. 82 (1976), no.Β 3, 494β496. MR 402054, DOI 10.1090/S0002-9904-1976-14071-2
- S. C. Power, Hankel operators on Hilbert space, Research Notes in Mathematics, vol. 64, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR 666699
- Donald Sarason, Generalized interpolation in $H^{\infty }$, Trans. Amer. Math. Soc. 127 (1967), 179β203. MR 208383, DOI 10.1090/S0002-9947-1967-0208383-8
- Shun Hua Sun, Bergman shift is not unitarily equivalent to a Toeplitz operator, Kexue Tongbao (English Ed.) 28 (1983), no.Β 8, 1027β1030. MR 763833
- Shun Hua Sun, On Toeplitz operators in the $\theta$-class, Sci. Sinica Ser. A 28 (1985), no.Β 3, 235β241. MR 794649
Bibliographic Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 103 (1988), 809-812
- MSC: Primary 47B35; Secondary 47B20
- DOI: https://doi.org/10.1090/S0002-9939-1988-0947663-4
- MathSciNet review: 947663