Weighted Join Operators on Directed Trees

A rooted directed tree $\mathscr T=(V, E)$ with can be extended to a directed graph $\mathscr T_\infty=(V_\infty, E_\infty)$ by adding a vertex $\infty$ to $V$ and declaring each vertex in $V$ as a parent of $\infty.$ One may associate with the extended directed tree a family of semigroup structures $\sqcup_{b}$ with extreme ends being induced by the join operation $\sqcup$ and the meet operation $\sqcap$. Each semigroup structure among these leads to a family of densely defined linear operators $W^{b}_{\lambda_u}$ acting on $\ell^2(V),$ which we refer to as weighted join operators at a given base point $b \in V_{\infty}$ with prescribed vertex $u \in V$. The extreme ends of this family are weighted join operators $W^{\mathsf{root}}_{\lambda_u}$ and weighted meet operators $W^{\infty}_{\lambda_u}$. In this paper, we systematically study these operators. We also present a more involved counter-part of weighted join operators on rootless directed trees. In both cases, the class of weighted join operators overlaps with the well-studied classes of complex Jordan operators and $n$-symmetric operators. An important half of this paper is devoted to the study of rank one extensions $W_{f, g}$ of weighted join operators, where $f \in \ell^2(V)$ and $g : V \to \mathbb C$ is unspecified. Unlike weighted join operators, these operators are not necessarily closed. We provide a couple of compatibility conditions involving the weight system $\lambda_u$ and $g$ to ensure closedness of $W_{f, g}$. We discuss the role of the Gelfand-triplet in the realization of the Hilbert space adjoint of $W_{f, g}$. Further, we describe various spectral parts of $W_{f, g}$ in terms of the weight system and the tree data. We also provide sufficient conditions for $W_{f, g}$ to be a sectorial operator. In case $\mathscr T$ is leafless, we characterize rank one extensions $W_{f, g}$, which admit compact resolvent.