{
  "id": "1806.08495",
  "version": "v1",
  "published": "2018-06-22T05:05:24.000Z",
  "updated": "2018-06-22T05:05:24.000Z",
  "title": "Weighted Join Operators on Directed Trees",
  "authors": [
    "Sameer Chavan",
    "Rajeev Gupta",
    "Kalyan B. Sinha"
  ],
  "comment": "90 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-22T05:05:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B37",
      "47B15",
      "47B20",
      "05C20",
      "47H06"
    ],
    "keywords": [
      "directed tree",
      "extreme ends",
      "weight system",
      "semigroup structure",
      "hilbert space adjoint"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 90,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}