Disintegration of Measures and Contractive 2-Variable Weighted Shifts

Abstract

In this paper we give a new proof of the existence of disintegration measures using the Hausdorff Moment Problem on a Borel measurable space X × Y , where X ≡ Y is the unit interval. Using this new tool, we can give an abstract solution, moreover, and a concrete necessary condition for the Lifting Problem for contractive 2-variable weighted shifts. In addition, we have a new, computable, and sufficient condition for the Lifting Problem for 2-variable weighted shifts, and an improved version of the Curto-Muhly-Xia conjecture [8] for 2-variable weighted shifts. Mathematics Subject Classification (2000). Primary 47B20, 47B37, 47A13, 28A50; Secondary 44A60, 47-04, 47A20.

Schur product techniques for commuting multivariable weighted shifts

“…In the case when T is subnormal, the Berger measure μ of T is given by Although the lemma below is a known result (see [20]), we give the complete proof for the reader's convenience. …”

“…It is well known that the commutativity of the pair is necessary but not sufficient [1,[15][16][17], and it has recently been shown that the joint hyponormality of the pair is necessary but not sufficient [10]. In previous work we gave abstract solutions ([6, Theorem 3.1], [20,Theorem 2.7]) and concrete necessary conditions, albeit not sufficient, for the lifting ( [11,Theorem 3.3], [20,Theorem 2.10]) for 2-variable weighted shifts. In this paper we study the hyponormality and subnormality of 2-variable weighted shifts using the Schur product techniques in matrices.…”

“…The class in Theorem 2. 8 contains the examples in [10,Section 5], [11,Corollary 4.7 and Proposition 4.10], [12,Remark 3.5], [13,Theorem 3.10] and [20,Example 2.12]. Finally, in Section 3, we give a complete characterization of hyponormality and subnormality in the class FC of flat, contractive 2-variable weighted shifts T ∈ H 0 with the condition W β = b, where b > 0 is given (see Fig.…”

Weak and Quadratic Hyponormality of 2-Variable Weighted Shifts and Their Examples

“…, n ( [8]). Because the structure of ntuple operators can be extended from the study of 2-tuple operators, many operator theorists have concentrated their studies to the structure of n-tuple operators (see [3], [4], [5], [6], [7], [9], [10], [11], [12], [13], etc.). Curto, Lee and Yoon considered the properties (such as, hyponormality, subnormality, flatness, etc.)…”

Backward Extension of Subnormal 2-Variable Weighted Shifts

“…In [10], Curto, Muhly and Xia conjectured that if T = (T 1 , T 2 ) is a pair of commuting subnormal operators on H, then the hyponormal condition of T is enough for the subnormality of T. In many recent papers containing ( [6], [7], [8][11], [12], [18]), the conjecture was showed negatively. Example 2.9 is also a simple new counterexample for the Curto-Muhly-Xia conjecture.…”