Subnormality of Powers of Multivariable Weighted Shifts. (28th November 2020) - Lee, Sang Hoon; Lee, Woo Young; Yoon, Jasang
Abstract : Given a pair T ≡ T 1, T 2 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 ≡ N 1, N 2 of normal extensions of T 1 and T 2 ; 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 = W α, β ≔ T 1, T 2 is subnormal, then T is subnormal if and only if a power T m, n ≔ T 1 m, T 2 n 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.