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

Abstract

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 $\{e_{(k_1,k_2)}{\}}_{k_1,k_2\ge 0}$ . \ 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 $k_1,k_2\ge 1$; 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 ${\mathbf{T}}^{(m,n)}:=(T_1^m,T_2^n)$ ($m,n\ge 1$). \

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References in corpus (2)

• Propagation phenomena for hyponormal 2-variable weighted shifts
• Reconstruction of the Berger measure when the core is of tensor form

Cited by in corpus (1)

• Aluthge transforms of 2-variable weighted shifts