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On complex symmetric weighted shifts. II

Piotr Budzyński

TL;DR

This work analyzes complex symmetry of weighted shifts on directed trees, extending classical truncated shifts. It develops a decomposition-based approach to identify when a weighted shift $S_{\boldsymbol \lambda}$ is $C$-symmetric for a conjugation $C$, with explicit necessary and sufficient weight-relations on two finite-depth tree models $\mathscr T_{\kappa,\theta}$ and $\mathscr T^2_\kappa$. The results show that complex symmetry depends on both the weight sequence and the underlying tree topology, and include concrete finite-tree examples illustrating when symmetry holds or fails. The paper further explores infinite-tree scenarios, presenting a broom-like depth-1 example where symmetry can occur under fast weight decay, and a depth-2 example where complex selfadjointness is impossible and the existence of symmetry remains open.

Abstract

Assorted weighted shifts over finite rooted directed trees are studied. Their complex symmetry is characterized.

On complex symmetric weighted shifts. II

TL;DR

This work analyzes complex symmetry of weighted shifts on directed trees, extending classical truncated shifts. It develops a decomposition-based approach to identify when a weighted shift is -symmetric for a conjugation , with explicit necessary and sufficient weight-relations on two finite-depth tree models and . The results show that complex symmetry depends on both the weight sequence and the underlying tree topology, and include concrete finite-tree examples illustrating when symmetry holds or fails. The paper further explores infinite-tree scenarios, presenting a broom-like depth-1 example where symmetry can occur under fast weight decay, and a depth-2 example where complex selfadjointness is impossible and the existence of symmetry remains open.

Abstract

Assorted weighted shifts over finite rooted directed trees are studied. Their complex symmetry is characterized.

Paper Structure

This paper contains 4 sections, 2 theorems, 49 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Two examples
  3. Complex symmetry over ${\mathscr T}_{\kappa,\theta}$ and ${\mathscr T}^2_\kappa$
  4. Infinite trees - two examples

Key Result

Theorem 3

Let $\kappa\in \mathbb Z_+$ and $\theta \in\mathbb N$. Let $\boldsymbol \lambda=\{\lambda_v\}_{V_{\kappa,\theta}^\circ}\subseteq \mathbb C\setminus\{0\}$ satisfy $\lambda_{1,j}=\lambda_{2,j}=:\lambda_j$, $j\in J_\theta$. Then the weighted shift operator $S_{\boldsymbol \lambda}$ on a directed tree $

Theorems & Definitions (8)

  • Example 1
  • Example 2
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Example 5
  • Example 6