Hyponormality of Toeplitz operators

Cowen, Carl C.

doi:10.1090/S0002-9939-1988-0947663-4

Hyponormality of Toeplitz operators
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Proc. Amer. Math. Soc. 103 (1988), 809-812 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. 3, 597–604. MR 428097
Ichiro Amemiya, Takashi Ito, and Tin Kin Wong, On quasinormal Toeplitz operators, Proc. Amer. Math. Soc. 50 (1975), 254–258. 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), 89–102. 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), 216–220. MR 749683
Carl C. Cowen, More subnormal Toeplitz operators, J. Reine Angew. Math. 367 (1986), 215–219. MR 839133, DOI 10.1515/crll.1986.367.215

Hyponormal and subnormal Toeplitz operators

P. R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76 (1970), 887–933. 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. 4, 529–564. MR 555777, DOI 10.1007/BF01691076
Takashi Itô and Tin Kin Wong, Subnormality and quasinormality of Toeplitz operators, Proc. Amer. Math. Soc. 34 (1972), 157–164. MR 303334, DOI 10.1090/S0002-9939-1972-0303334-7

Hyponormal Toeplitz operators and weighted shifts

Donald E. Marshall, Blaschke products generate $H^{\infty }$, Bull. Amer. Math. Soc. 82 (1976), no. 3, 494–496. 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), 179–203. 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. 8, 1027–1030. MR 763833
Shun Hua Sun, On Toeplitz operators in the $\theta$-class, Sci. Sinica Ser. A 28 (1985), no. 3, 235–241. MR 794649