A hyponormal weighted shift on a directed tree whose square has trivial domain. (English) Zbl 1301.47043
Motivated by the question of whether there exists a subnormal weighted shift on a direct tree with nonzero weights whose square has trivial domain, the authors prove that, up to isomorphism, there are only two directed trees that admit an injective hyponormal weighted shift with nonzero weights whose square has trivial domain. These are precisely those enumerable (i.e., countably infinite) directed trees, one with root, the other without, whose every vertex has an enumerable set of successors. They also provide an example of a nonzero hyponormal composition operator in an \(L^2\)-space over a \(\sigma\)-finite space whose square has trivial domain.
Reviewer: Abdellatif Bourhim (Syracuse)
MSC:
| 47B37 | Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) |
| 47B20 | Subnormal operators, hyponormal operators, etc. |
| 47A05 | General (adjoints, conjugates, products, inverses, domains, ranges, etc.) |
Keywords:
directed tree; weighted shift on a directed tree; hyponormal operator; trivial domain of squareReferences:
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