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

TL;DR

This paper investigates the Lifting Problem for commuting subnormal 2-variable weighted shifts, showing that subnormality for arbitrary powers is equivalent to subnormality of the original shift when the core has tensor form.

Contribution

It establishes a characterization of the Lifting Problem's solvability for shifts with tensor core by examining their powers, providing a new criterion for subnormality.

Findings

LPCS is solvable for a shift if and only if it is solvable for all its powers.

The core of the shift being of tensor form is crucial for the main result.

The result applies to shifts with a tensor-form core, linking subnormality of powers to the original shift.

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 $\{e_{(k_1,k_2)}\}_{k_1,k_2 \geq 0}$ . \ 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 $\mathbf{T}^{(m,n)}:=(T^m_1,T^n_2)$ ($m,n\geq 1$). \