Subnormality of Powers of Multivariable Weighted Shifts

Abstract

Given a pair $\mathbf{T}\equiv \left(T_1,T_2\right)$ of commuting subnormal Hilbert space operators, the Lifting Problem for Commuting Subnormals (LPCS) asks for necessary and sufficient conditions for the existence of a commuting pair $\mathbf{N}\equiv \left(N_1,N_2\right)$ of normal extensions of $T_1$ and $T_2$ ; in other words, $\mathbf{T}$ is a subnormal pair. The LPCS is a longstanding open problem in the operator theory. In this paper, we consider the LPCS of a class of powers of $2$ -variable weighted shifts. Our main theorem states that if a “corner” of a 2-variable weighted shift $\mathbf{T}=W_{\left(\alpha ,\beta \right)}≔\left(T_1,T_2\right)$ is subnormal, then $\mathbf{T}$ is subnormal if and only if a power ${\mathbf{T}}^{\left(m,n\right)}≔\left(T_1^m,T_2^n\right)$ is subnormal for some $m,n\ge 1$ . As a corollary, we have that if $\mathbf{T}$ is a 2-variable weighted shift having a tensor core or a diagonal core, then $\mathbf{T}$ is subnormal if and only if a power of $\mathbf{T}$ is subnormal.