A new characterization of subnormality for a class of 2-variable weighted shifts with 1-atomic core

Given a pair T≡(T1,T2) of commuting subnormal Hilbert space operators, the Lifting Problem for Commuting Subnormals (LPCS) calls for necessary and sufficient conditions for the existence of a commuting pair N≡(N1,N2) of normal extensions of T1 and T2. This is an old problem in operator theory. The aim of this paper is to study LPCS. There are three well-known subnormal characterizations for operators: the Berger Theorem, the Bram-Halmos characterization, and Franks' result. In our paper, we study a new subnormal characterization which is related to these three well-known ones for a class of 2-variable weighted shifts. Thus, we can provide a large nontrivial class of 2-variable weighted shifts in which k-hyponormal (some k≥1) and subnormal are equal and the class is invariant under the action (h,ℓ)↦T(h,ℓ):=(T1h,T2ℓ) (h,ℓ≥1).