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Completion problem for extension of m- isometric weighted composition operators on directed graphs

V. Devadas, T. Prasad, E. Shine Lal

TL;DR

The paper addresses how to extend finite weight data to $k$-quasi-$m$-isometric operators—specifically weighted shifts, composition operators, and weighted composition operators—on directed graphs with a single circuit and multiple branches. It develops both unweighted and DSP-based (discrete spectral) frameworks to translate quasi-isometric conditions into polynomial interpolation problems for moment sequences and Radon-Nikodym data, yielding explicit existence and construction criteria. Key contributions include general completion theorems for cases $l \le k+m-2$ and $l > k+m-2$, explicit characterizations of low-circuit $1$-quasi-$3$- and $1$-quasi-$2$-isometries on graphs, and DSP-based procedures to realize and perturb $k$-quasi-$m$-isometric weighted composition operators. Collectively, the work extends completion theory to graph-based operator classes and provides concrete, interpolation-driven methods for constructing and stabilizing such completions.

Abstract

In this paper, we discuss k-quasi-m-isometric completion problem of unilateral weighted shifts and composition operators on directed graphs with one circuit and more than one branching vertex.

Completion problem for extension of m- isometric weighted composition operators on directed graphs

TL;DR

The paper addresses how to extend finite weight data to -quasi--isometric operators—specifically weighted shifts, composition operators, and weighted composition operators—on directed graphs with a single circuit and multiple branches. It develops both unweighted and DSP-based (discrete spectral) frameworks to translate quasi-isometric conditions into polynomial interpolation problems for moment sequences and Radon-Nikodym data, yielding explicit existence and construction criteria. Key contributions include general completion theorems for cases and , explicit characterizations of low-circuit -quasi-- and -quasi--isometries on graphs, and DSP-based procedures to realize and perturb -quasi--isometric weighted composition operators. Collectively, the work extends completion theory to graph-based operator classes and provides concrete, interpolation-driven methods for constructing and stabilizing such completions.

Abstract

In this paper, we discuss k-quasi-m-isometric completion problem of unilateral weighted shifts and composition operators on directed graphs with one circuit and more than one branching vertex.
Paper Structure (3 sections, 10 theorems, 63 equations, 1 figure)

This paper contains 3 sections, 10 theorems, 63 equations, 1 figure.

Table of Contents

  1. Introduction and Preliminaries
  2. $k$-quasi-$m$-isometric completion problem for weighted shift and composition operators
  3. $k$-quasi-$m$-isometry completion problem for Weighted composition operators.

Key Result

Lemma 2.1

ZJJK Let $l \in \mathbb{Z}^+$ and $\{ b_n \}_{n=0}^l\subset (0, \infty)$. Then, there exists $c \in (0, \infty)$ such that for every $t \in [c, \infty)$, there exists a polynomial $w_t(x) \in \mathbb{R}[x]$ of degree $l+1$ such that: $w_t(n) = b_n,~~\text{for}~~ n \in J_{[0, l]},$$w_t(l+1) = t,$ and

Figures (1)

  • Figure 1: Directed graph with one circuit and more than one branching vertex

Theorems & Definitions (22)

  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Example 2.3
  • Example 2.4
  • Remark 2.5
  • Theorem 2.6
  • proof
  • Theorem 2.7
  • proof
  • ...and 12 more