Abstract
For a given directed tree and weights attached to a subtree, the completion problem is to determine if these weights may be completed in a way to obtain a bounded weighted shift on the whole tree, which further satisfies additional conditions. In this paper we consider subnormal and completely hyperexpansive completion problem for weighted shifts on directed trees with one branching point. We develop new results on backward extensions of truncated moment sequences and, exploiting these results, we obtain a characterization of existence of such a completion.
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