Hyponormality of Toeplitz Operators Carl C. Cowen * Proc. Amer. Math. Soc. 103(1988) 809-812. Abstract For ϕ in L ∞ (∂D), let ϕ = f + g where f and g are in H 2 . In this note, it is shown that the Toeplitz operator T ϕ is hyponormal if and only if g = c + T h f for some constant c and some function h in H ∞ (∂D) with khk ∞ ≤ 1. For ϕ in L ∞ (∂D), the Toeplitz operator T ϕ is the operator on H 2 of the unit disk D given by T ϕ u = P ϕu where P is the orthogonal projection of L 2 (∂D) onto H 2 . An operator A is called hyponormal if its self-commutator A * A - AA * is positive. The goal of this paper is to characterize hyponormal Toeplitz operators. Brown and Halmos began the systematic study of the algebraic prop- erties of Toeplitz operators and showed, [3, page 98], that T ϕ is normal if and only if ϕ = α + βρ where α and β are complex numbers and ρ is a real valued function in L ∞ . There are many results concerning hyponormality of Toeplitz operators in the literature and properties of hyponormal Toeplitz operators have played an important role in work on Halmos’s Problem 5, [7], “Is every subnormal Toeplitz operator either normal or analytic?” but a characterization has been lacking. (For references, see the bibliography; [6] surveys much of the literature.) 0 1980 Mathematics Subject Classification (1985 Revision). Primary 47B35, 47B20. 0 Key Words and Phrases. Toeplitz operator, subnormal operator. * Supported in part by National Science Foundation Grant DMS 87-10006. 1