Subnormality of Powers of Multivariable Weighted Shifts

Sang Hoon Lee,Woo Young Lee,Jasang Yoon

doi:10.1155/2020/5678795

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.

Highlights

For a Hilbert space operator, a subnormal operator means an operator admitting a normal extension, i.e., an extension which is a normal operator
In [5, 6], we have examined the above results for the class of 2-variable weighted shifts T = Wðα,βÞ
For the class of 2-variable weighted shifts T = Wðα,βÞ with a core of tensor form, denoted T C [5], or with a core of diagonal form, denoted DC [6], we have shown that if T = Wðα,βÞ ∈ T C ∪ DC, the following statements are equivalent: (a) T is subnormal (b) Tðm,nÞ is subnormal for all m, n ≥ 1 (c) Tðm,nÞ is subnormal for some m, n ≥ 1

Summary

Introduction

For a Hilbert space operator, a subnormal operator means an operator admitting a normal extension, i.e., an extension which is a normal operator. The reason why we take 2-variable weighted shifts for examining the subnormality of powers for pairs of operators is that 2-variable weighted shifts play an important role in detecting properties such as subnormality, via the Lambert-Lubin Criterion ([9, 10]): a commuting pair ðT1, T2Þ of injective operators acting on a Hilbert space H admits a commuting normal extension if and only if for every nonzero vector x ∈ H , the 2-variable weighted shift with weights αði,jÞ ≔ TTi1+i11TT2j2jxx and βði,jÞ ≔ TTi1i1TT2j+2j1xx , ð3Þ has a normal extension.

Preliminaries and the Main Theorem

The Proof of the Main Theorem