Subnormality for arbitrary powers of 2-variable weighted shifts whose restrictions to a large invariant subspace are tensor products

AbstractThe Lifting Problem for Commuting Subnormals (LPCS) asks for necessary and sufficient conditions for a pair of subnormal operators on Hilbert space to admit commuting normal extensions. We study LPCS within the class of commuting 2-variable weighted shifts T≡(T1,T2) with subnormal components T1 and T2, acting on the Hilbert space ℓ2(Z+2) with canonical orthonormal basis {e(k1,k2)}k1,k2⩾0. The core of a commuting 2-variable weighted shift T, c(T), is the restriction of T to the invariant subspace generated by all vectors e(k1,k2) with k1,k2⩾1; we say that c(T) is of tensor form if it is unitarily equivalent to a shift of the form (I⊗Wα,Wβ⊗I), where Wα and Wβ are subnormal unilateral weighted shifts. Given a 2-variable weighted shift T whose core is of tensor form, we prove that LPCS is solvable for T if and only if LPCS is solvable for any power T(m,n):=(T1m,T2n) (m,n⩾1)