Journal article
When is hyponormality for 2-variable weighted shifts invariant under powers?
Indiana University mathematics journal, Vol.60(3), pp.997-1032
2011
DOI: 10.1512/iumj.2011.60.4303
Abstract
For 2-variable weighted shifts W_{(\alpha,\beta)}(T_1, T_2) we study the invariance of (joint) k- hyponormality under the action (h,\ell) -> W_{(\alpha,\beta)}^{(h,\ell)}(T_1, T_2):=(T_1^k,T_2^{\ell}) (h,\ell >=1). We show that for every k >= 1 there exists W_{(\alpha,\beta)}(T_1, T_2) such that W_{(\alpha,\beta)}^{(h,\ell)}(T_1, T_2) is k-hyponormal (all h>=2,\ell>=1) but W_{(\alpha,\beta)}(T_1, T_2) 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.
Details
- Title: Subtitle
- When is hyponormality for 2-variable weighted shifts invariant under powers?
- Creators
- Raúl E CurtoJasang Yoon
- Resource Type
- Journal article
- Publication Details
- Indiana University mathematics journal, Vol.60(3), pp.997-1032
- DOI
- 10.1512/iumj.2011.60.4303
- ISSN
- 0022-2518
- eISSN
- 1943-5258
- Language
- English
- Date published
- 2011
- Academic Unit
- Mathematics
- Record Identifier
- 9983985829902771
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