Abstract. In this note we are concerned with the hyponormality of Toeplitz operators T ϕ with polynomial symbols ϕ =ḡ + f (f, g ∈ H ∞ (T)) when g divides f .
Suppose Tϕ is a 2-hyponormal Toeplitz operator whose selfcommutator has rank n ≥ 1. If Hφ ker [T * ϕ , Tϕ] contains a vector en in a canonical orthonormal basis {e k } k∈Z + of H 2 (T), then ϕ should be an analytic function of the form ϕ = qh, where q is a finite Blaschke product of degree at most n and h is an outer function.
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