SOLUTION OF THE QUADRATICALLY HYPONORMAL COMPLETION PROBLEM Ra´ ul E. Curto and Woo Young Lee For m ≥ 1, let α : α 0 < ··· <α m be a collection of (m + 1) positive weights. The Quadratically Hyponormal Completion Problem seeks necessary and sufficient conditions on α to guarantee the existence of a quadratically hyponormal unilateral weighted shift W with α as initial segment of weights. We prove that α admits a quadratically hyponormal completion if and only if the self-adjoint m × m matrix D m-1 (s) := q 0 ¯ r 0 0 ... 0 0 r 0 q 1 ¯ r 1 ... 0 0 0 r 1 q 2 ... 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 ... q m-2 ¯ r m-2 0 0 0 ... r m-2 q m-1 is positive and invertible, where q k := u k + |s| 2 v k , r k := s √ w k , u k := α 2 k - α 2 k-1 , v k := α 2 k α 2 k+1 - α 2 k-1 α 2 k-2 , w k := α 2 k (α 2 k+1 - α 2 k-1 ) 2 , and, for notational conve- nience, α -2 = α -1 = 0. As a particular case, this result shows that a collection of four positive numbers α 0 <α 1 <α 2 <α 3 always admits a quadratically hyponor- mal completion. This provides a new qualitative criterion to distinguish quadratic hyponormality from 2-hyponormality. 1. Introduction Let H be a complex Hilbert space and let L(H) be the set of bounded linear operators on 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. The Bram-Halmos criterion for subnormality states that an operator T is subnormal if and only if ∑ i,j (T i x j ,T j x i ) ≥ 0 for all finite collections x 0 ,x 1 , ··· ,x k ∈H ([2],[5, II.1.9]). It is easy to see that this is equivalent to the following positivity test: (1.1)     I T * ... T *k T T * T ... T *k T . . . . . . . . . . . . T k T * T k ... T *k T k     ≥ 0 (all k ≥ 1). 1991 Mathematics Subject Classification. Primary 47B20, 47B35, 47B37; Secondary 47-04, 47A20, 47A57. Key words and phrases. Weighted shifts, propagations, subnormal, k-hyponormal, quadrati- cally hyponormal, completions. The work of the first-named author was partially supported by NSF research grants DMS- 9800931 and DMS-0099357. The work of the second-named author was partially supported by the Brain Korea 21 Project. Typeset by A M S-T E X 1