When is hyponormality for 2-variable weighted shifts invariant under powers?

Authors

Raul E. Curto, University of Iowa
Jasang Yoon, The University of Texas Rio Grande ValleyFollow

Document Type

Article

Publication Date

4-11-2011

Abstract

Abstract. For 2-variable weighted shifts W(α,β) ≡ (T1, T2) we study the invariance of (joint) khyponormality under the action (h, ℓ) 7→ W (h,ℓ) (α,β) := (T h 1 , T ℓ 2 ) (h, ℓ ≥ 1). We show that for every k ≥ 1 there exists W(α,β) such that W (h,ℓ) (α,β) is k-hyponormal (all h ≥ 2, ℓ ≥ 1) but W(α,β) is not k-hyponormal. On the positive side, for a class of 2-variable weighted shifts with tensor core we find a computable necessary condition for invariance. Next, we exhibit a large nontrivial class for which hyponormality is indeed invariant under all powers; moreover, for this class 2-hyponormality automatically implies subnormality. Our results partially depend on new formulas for the determinant of generalized Hilbert matrices and on criteria for their positive semi-definiteness.

Comments

Original published version available at https://www.jstor.org/stable/24903826

Recommended Citation

Curto, Raúl E., and Jasang Yoon. “When Is Hyponormality for 2-Variable Weighted Shifts Invariant under Powers?” Indiana University Mathematics Journal, vol. 60, no. 3, 2011, pp. 997–1032., www.jstor.org/stable/24903826.

First Page

997

Last Page

1032

Publication Title

Indiana University Mathematics Journal