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Backward Extensions of Weighted Shifts on Directed Trees

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Abstract

The weighted shifts are long known and form an important class of operators. One of generalisations of this class are weighted shifts on directed trees, where the linear order of coordinates in \(\ell ^2\) is replaced by a more involved graph structure. In this paper we focus on the question of joint backward extending of a given family of weighted shifts on directed trees to a weighted shift on an enveloping directed tree that preserves subnormality or power hyponormality of considered operators. One of the main results shows that the existence of such a “joint backward extension” for a family of weighted shifts on directed trees depends only on the possibility of backward extending of single weighted shifts that are members of the family. We introduce a generalised framework of weighted shifts on directed forests (disjoint families of directed trees) which seems to be more convenient to work with. A characterisation of leafless directed forests on which all hyponormal weighted shifts are power hyponormal is given.

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Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Notes

  1. According to [19, Corollary 1] (cf. Theorem 3.7), there is no loss of generality in assuming that the weights are non-negative.

  2. Except for the weights attached to the roots, which equal 0 by definition.

  3. Note that if a root has no children (it is not considered as a child of itself) it still cannot be called a ‘leaf’.

  4. Note that “\(\mathtt {h}_{k}(\omega )\)” is computed for \(S_ {\varvec{{\tilde{\lambda }}}} \) on the directed tree , while “\(\mathtt {h}_{k}\bigl (\omega _{({\mathcal {T}}_j)_{\langle 1\rangle }} \bigr )\)” refers to the weighted shift \(S_{{\varvec{\lambda }}_j}\) on \({{\mathcal {T}}_j}_{\langle 1\rangle }\).

  5. We follow the convention that \(\frac{1}{0}=\infty \). In particular \(\int _0^\infty \frac{1}{x^k} \mathrm {\,d}\mu (x) <\infty \) implies \(\mu (\{0\})=0\).

  6. The term depth appears here without a formal definition, as it is not necessary in the statement of the theorem. Figures 5 and 6 should give an intuition.

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Acknowledgements

I would like to thank my supervisor, prof. Jan Stochel, for all his guidance while working on this paper.

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  1. Instytut Matematyki, Wydział Matematyki i Informatyki, Uniwersytet Jagielloński, ul. Łojasiewicza 6, 30-348, Kraków, Poland

    Piotr Pikul

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Pikul, P. Backward Extensions of Weighted Shifts on Directed Trees. Integr. Equ. Oper. Theory 94, 26 (2022). https://doi.org/10.1007/s00020-022-02704-5

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