A hyponormal weighted shift on a directed tree whose square has trivial domain

Jabłoński, Zenon; Jung, Il Bong; Stochel, Jan

doi:10.1090/S0002-9939-2014-12112-5

A hyponormal weighted shift on a directed tree whose square has trivial domain
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by Zenon Jan Jabłoński, Il Bong Jung and Jan Stochel PDF

Proc. Amer. Math. Soc. 142 (2014), 3109-3116 Request permission

Abstract:

It is proved that, up to isomorphism, there are only two directed trees that admit a 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. An example of a nonzero hyponormal composition operator in an $L^2$-space whose square has trivial domain is established.

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