Subnormality for arbitrary powers of 2-variable weighted shifts whose restrictions to a large invariant subspace are tensor products
نویسندگان
چکیده
The 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 l (Z+) 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 := (T 1 , T n 2 ) (m,n ≥ 1).
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