Subnormality of Powers of Multivariable Weighted Shifts

Abstract EN

Given a pair T≡T1,T2 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 N≡N1,N2 of normal extensions of T1 and T2; in other words, 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 T=Wα,β≔T1,T2 is subnormal, then T is subnormal if and only if a power Tm,n≔T1m,T2n is subnormal for some m,n≥1.

As a corollary, we have that if T is a 2-variable weighted shift having a tensor core or a diagonal core, then T is subnormal if and only if a power of T is subnormal.