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

2011/04/18 by Raúl E. Curto, Raul Curto, Curto, Raul +2
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Holomorphic and Operator Theory #math.FA #msc:28A50 #msc:44A60 #msc:47-04 #msc:47A13 #msc:47A20 #msc:47B20 #msc:47B37

paper · pdf · doi:10.48550/arxiv.1104.3604

arxiv created 2011/04/18 · arxiv updated 2011/04/20

Abstract

For 2-variable weighted shifts W_(α,β)(T₁, T₂) we study the invariance of (joint) k- hyponormality under the action (h,ℓ) -> W_(α,β)^(h,ℓ)(T₁, T₂):=(T₁^k,T₂^ℓ) (h,ℓ >=1). We show that for every k >= 1 there exists W_(α,β)(T₁, T₂) such that W_(α,β)^(h,ℓ)(T₁, T₂) is k-hyponormal (all h>=2,ℓ>=1) but W_(α,β)(T₁, T₂) 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.

Related

• k-hyponormality of powers of weighted shifts via Schur products
• Propagation phenomena for hyponormal 2-variable weighted shifts
• Spectral Characterization of Hyponormal Weighted Shifts
• Polynomial embeddings of unilateral weighted shifts into 2-variable weighted shifts
• Semi-cubic hyponormality of weighted shifts with stampfli recursive tail