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On operator fields in the upper triangular Toeplitz form

M. M. Chernin, A. Yu. Konyaev

TL;DR

This work addresses the problem of describing all coordinate transformations that preserve the upper triangular Toeplitz form of an operator field $L$, linking the problem to Nijenhuis geometry and Haantjes theory. It provides a PDE-based characterization (via $d g_i$) of when such Toeplitz operators are Nijenhuis, and then offers an explicit constructive framework: (i) a $P,Q$–based functional calculus yielding all Nijenhuis Toeplitz operators with $g_{n-1}\neq0$, (ii) a linear system for coordinate changes that bring $L$ to the Jordan form $J$, and (iii) an integration-based algorithm to solve the prescribed system, with concrete dimension four examples. The results unify Toeplitz-structure preservation with Nijenhuis geometry, delivering practical procedures to classify and compute preserving transformations and revealing how these transformations act in applications across integrable systems and complex geometric settings. Overall, the paper provides both a theoretical framework and concrete algorithms for transforming and classifying Toeplitz-structured operator fields in high dimensions.

Abstract

In this work, we solve the fundamental problem of describing the coordinate transformations that preserve the upper triangular Toeplitz form of the given operator field. Surprisingly, this problem is closely related to the description of all Nijenhuis operators in the same form. This description, as well as the formulas for the aforementioned coordinate transformations, are given by the implicit formulas involving matrix-valued functions.

On operator fields in the upper triangular Toeplitz form

TL;DR

This work addresses the problem of describing all coordinate transformations that preserve the upper triangular Toeplitz form of an operator field , linking the problem to Nijenhuis geometry and Haantjes theory. It provides a PDE-based characterization (via ) of when such Toeplitz operators are Nijenhuis, and then offers an explicit constructive framework: (i) a –based functional calculus yielding all Nijenhuis Toeplitz operators with , (ii) a linear system for coordinate changes that bring to the Jordan form , and (iii) an integration-based algorithm to solve the prescribed system, with concrete dimension four examples. The results unify Toeplitz-structure preservation with Nijenhuis geometry, delivering practical procedures to classify and compute preserving transformations and revealing how these transformations act in applications across integrable systems and complex geometric settings. Overall, the paper provides both a theoretical framework and concrete algorithms for transforming and classifying Toeplitz-structured operator fields in high dimensions.

Abstract

In this work, we solve the fundamental problem of describing the coordinate transformations that preserve the upper triangular Toeplitz form of the given operator field. Surprisingly, this problem is closely related to the description of all Nijenhuis operators in the same form. This description, as well as the formulas for the aforementioned coordinate transformations, are given by the implicit formulas involving matrix-valued functions.
Paper Structure (6 sections, 11 theorems, 70 equations)

This paper contains 6 sections, 11 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof of Theorem \ref{['t1']}
  4. Proof of Theorem \ref{['t2']}
  5. Proof of Theorem \ref{['t3']} and Remark \ref{['r1']}
  6. Proof of Theorem \ref{['t4']}

Key Result

Theorem 1

Fix the coordinates $u^1, \dots, u^n$ and assume that the operator field $L$ is in the upper triangular Toeplitz form. The following (equivalent) conditions on $g_i, i = 1, \dots, n$ are sufficient for $L$ to be Nijenhuis: If $g_{n - 1} \neq 0$, then the conditions above are also necessary.

Theorems & Definitions (22)

  • Theorem 1
  • Example 2.1
  • Theorem 2
  • Example 2.2
  • Theorem 3
  • Remark 2.1
  • Theorem 4
  • Example 2.3
  • Lemma 3.1
  • proof
  • ...and 12 more