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

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

2011-10-30

arxiv.org/abs/1110.6611v1

Journal of Functional Analysis 262 (2012) 569-583

Mathematics

Functional Analysis

article in press

Scientific paper

10.1016/j.jfa.2011.09.024

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 $\mathbf{T} \equiv (T_1,T_2)$ with subnormal components $T_1$ and $T_2$, acting on the Hilbert space $\ell ^2(\mathbb{Z}^2_+)$ with canonical orthonormal basis $\{e_{(k_1,k_2)}\}_{k_1,k_2 \geq 0}$ . \ The \textit{core} of a commuting 2-variable weighted shift $\mathbf{T}$, $c(\mathbf{T})$, is the restriction of $\mathbf{T}$ to the invariant subspace generated by all vectors $e_{(k_1,k_2)}$ with $k_1,k_2 \geq 1$; we say that $c(\mathbf{T})$ is of \textit{tensor form} if it is unitarily equivalent to a shift of the form $(I \otimes W_\alpha, W_\beta \otimes I)$, where $W_\alpha$ and $W_\beta$ are subnormal unilateral weighted shifts. \ Given a 2-variable weighted shift $\mathbf{T}$ whose core is of tensor form, we prove that LPCS is solvable for $\mathbf{T}$ if and only if LPCS is solvable for any power $\mathbf{T}^{(m,n)}:=(T^m_1,T^n_2)$ ($m,n\geq 1$). \

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