SUBNORMALITY AND 2-HYPONORMALITY FOR TOEPLITZ OPERATORS Ra´ ul E. Curto, 1 Sang Hoon Lee and Woo Young Lee 2 In this article we provide an example of a Toeplitz operator which is 2–hyponormal but not subnormal, and we consider 2-hyponormal Toeplitz operators with finite rank self-commutators. The present article concerns the gap between subnormality and 2-hyponormality for Toeplitz operators. We begin with a brief survey of research related to P.R. Halmos’s Problem 5 (cf. [Ha1],[Ha2]): (Prob 5) Is every subnormal Toeplitz operator either normal or analytic ? As we know, (Prob 5) was answered in the negative by C. Cowen and J. Long [CoL]. Directly connected with it is the following problem: (0.1) Which Toeplitz operators are subnormal ? Let H and K be complex Hilbert spaces, let L(H, K) be the set of bounded linear operators from H to K and write L(H) := L(H, H). An operator T ∈L(H) is said to be normal if T * T = TT * , hyponormal if T * T ≥ TT * , and subnormal if T = N | H , where N is normal on some Hilbert space K⊇H. If T is subnormal then T is also hyponormal. Recall that the Hilbert space L 2 (T) has a canonical orthonormal basis given by the trigonometric functions e n (z )= z n , for all n ∈ Z, and that the Hardy space H 2 (T) is the closed linear span of {e n : n =0, 1, ···}. An element f ∈ L 2 (T) is said to be analytic if f ∈ H 2 (T), and co-analytic if f ∈ L 2 (T) ⊖ H 2 (T). If P denotes the orthogonal projection from L 2 (T) to H 2 (T), then for every ϕ ∈ L ∞ (T) the operators T ϕ and H ϕ on H 2 (T) defined by T ϕ g := P (ϕg) and H ϕ (g) := (I − P )(ϕg) (g ∈ H 2 (T)) 1 Supported by NSF research grant DMS-9800931. 2 Supported by KOSEF research project No. R01-2000-00003.