k-HYPONORMALITY OF MULTIVARIABLE WEIGHTED SHIFTS

We characterize joint k-hyponormality for 2-variable weighted shifts. Using this characterization we construct a family of examples which establishes and illustrates the gap between k-hyponormality and (k+1)-hyponormality for each k ≥ 1. As a consequence, we obtain an abstract solution to the Lifting Problem for Commuting Subnormals. 1. Notation and Preliminaries The Lifting Problem for Commuting Subnormals asks for necessary and sufficient conditions for a pair of subnormal operators on Hilbert space to admit commuting normal extensions. It is well known that the commutativity of the pair is necessary but not sufficient ([Abr], [Lu1], [Lu2], [Lu3]), and it has recently been shown that the joint hyponormality of the pair is necessary but not sufficient [CuYo1]. In this paper we provide an abstract answer to the Lifting Problem, by stating and proving a multivariable analogue of the Bram-Halmos criterion for subnormality, and then showing concretely that no matter how k-hyponormal a pair might be, it may still fail to be subnormal. To do this, we obtain a matricial characterization of k-hyponormality for multivariable weighted shifts, which extends that found in [Cu1] for joint hyponormality. Let H be a complex Hilbert space and let B(H) denote the algebra of bounded linear operators on H. For S, T ∈ B(H) let [S, T ] := ST − TS. We say that an n -tuple T = (T1, · · · , Tn) of operators on H is (jointly) hyponormal if the operator matrix [T∗,T] := ⎛ ⎜⎜⎝ [T ∗ 1 , T1] [T ∗ 2 , T1] · · · [T ∗ n , T1] [T ∗ 1 , T2] [T ∗ 2 , T2] · · · [T ∗ n , T2] .. .. . . . .. [T ∗ 1 , Tn] [T ∗ 2 , Tn] · · · [T ∗ n , Tn] ⎞ ⎟⎟⎠ (1.1) is positive on the direct sum of n copies of H (cf. [Ath] , [CMX]). The n-tuple T is said to be normal if T is commuting and each Ti is normal, and T is subnormal if T is the restriction of a normal n-tuple to a common invariant subspace. Clearly, normal ⇒ subnormal ⇒ hyponormal. Moreover, the restriction of a hyponormal n-tuple to an invariant subspace is again hyponormal. The Bram-Halmos criterion states that an operator T ∈ B(H) is subnormal if and only if the k -tuple (T, T 2, · · · , T k) is hyponormal for all k ≥ 1. For α ≡ {αn}∞n=0 a bounded sequence of positive real numbers (called weights), let Wα : (Z+) → (Z+) be the associated unilateral weighted shift, defined by Wαen := αnen+1 (all n ≥ 0), where {en}∞n=0 is the canonical orthonormal basis in (Z+). The moments of α are given as γk ≡ γk(α) := { 1 if k = 0 α0 · · ·αk−1 if k > 0 } . 1991 Mathematics Subject Classification. Primary 47B20, 47B37, 47A13, 28A50; Secondary 44A60, 47-04, 47A20.