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Normal extensions escape from the class of weighted shifts on directed trees. (English) Zbl 1296.47029
In [Mem. Am. Math. Soc. 1017 (2012; Zbl 1248.47033)], the authors introduced an interesting new class of (not necessarily bounded) operators in Hilbert spaces, called weighted shifts on directed trees. These operators generalize the well known notion of weighted shift operators. In the present paper, the authors continue their study. They show that formally normal weighted shifts on directed trees are always bounded and normal. The question whether a normal extension of a subnormal weighted shift on a directed tree can be modeled as a weighted shift on some (possibly different) directed tree is answered as well. More precisely, they prove that a nonzero weighted shift $S_{\lambda}$ on a directed tree ${\mathcal T}$ with nonzero weights has a normal extension which is a weighted shift on a directed tree if and only if either ${\mathcal T}$ is isomorphic to ${\mathbb Z}$ and $S_{\lambda}$ is unitarily equivalent to a positive scalar multiple of the bilateral shift on $\ell^{2}({\mathbb Z})$, or ${\mathcal T}$ is isomorphic to ${\mathbb Z}_{+}$ and $S_{\lambda}$ is unitarily equivalent to a positive scalar multiple of a unilateral weighted shift on $\ell^{2}({\mathbb Z}_{+})$ with weights $\{ \vartheta,1,1,1,\ldots\}$, where $\vartheta \in (0,1]$.

MSC:
47B37Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47B20Subnormal operators, hyponormal operators, etc.
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References:
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