An Analytic Model for Left-Invertible Weighted Shifts on Directed Trees

Abstract

Let $\mathcal{T}$ be a rooted directed tree with finite branching index $k_{\mathcal{T}}$ and let $S_{\lambda }\in B(l^2(V))$ be a left-invertible weighted shift on $\mathcal{T}$. We show that $S_{\lambda }$ can be modelled as a multiplication operator ${\mathcal{M}}_z$ on a reproducing kernel Hilbert space $\mathcal{H}$ of $E$-valued holomorphic functions on a disc centered at the origin, where $E:=\ker S_{\lambda }^{\ast }$. The reproducing kernel associated with $\mathcal{H}$ is multi-diagonal and of bandwidth $k_{\mathcal{T}}.$ Moreover, $\mathcal{H}$ admits an orthonormal basis consisting of polynomials in $z$ with at most $k_{\mathcal{T}}+1$ non-zero coefficients. As one of the applications of this model, we give a complete spectral picture of $S_{\lambda }.$ Unlike the case $\dim E=1,$ the approximate point spectrum of $S_{\lambda }$ could be disconnected. We also obtain an analytic model for left-invertible weighted shifts on rootless directed trees with finite branching index.

References in corpus (1)

• Analytic structure of weighted shifts on directed trees

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