Subnormality of Powers of Multivariable Weighted Shifts

Given a pair 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 of normal extensions of and ; in other words, 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 -variable weighted shifts. Our main theorem states that if a “corner” of a 2-variable weighted shift is subnormal, then is subnormal if and only if a power is subnormal for some . As a corollary, we have that if is a 2-variable weighted shift having a tensor core or a diagonal core, then is subnormal if and only if a power of is subnormal