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A Combinatorial Formula for Recursive Operator Sequences and Applications

Raul E. Curto, Abderrazzak Ech-charyfy, Kaissar Idrissi, El Hassan Zerouali

Abstract

We study sequences of bounded operators \((T_n)_{n \ge 0}\) on a complex separable Hilbert space \(\mathcal{H}\) that satisfy a linear recurrence relation of the form $$ T_{n+r} = A_0 T_n + A_1 T_{n+1} + \cdots + A_{r-1} T_{n+r-1} \quad(\textrm{for all } n\ge 0), $$ where the coefficients \(A_0, A_1, \dots, A_{r-1}\) are pairwise commuting bounded operators on \(\mathcal{H}\). \ Such relations naturally arise in the context of the operator-valued moment problem, particularly in the study of flat extensions of block Hankel operators. \ Our first goal is to derive an explicit combinatorial formula for \(T_n\). As a concrete application, we provide an explicit expression for the powers of an operator-valued companion matrix. \ In the special case of scalar coefficients $A_k=a_kI_\mathcal{H}$, with $a_k\in\mathbb{R}$, we recover a Binet-type formula that allows the explicit computation of the powers and the exponential of algebraic operators in terms of Bell polynomials.

A Combinatorial Formula for Recursive Operator Sequences and Applications

Abstract

We study sequences of bounded operators \((T_n)_{n \ge 0}\) on a complex separable Hilbert space that satisfy a linear recurrence relation of the form where the coefficients are pairwise commuting bounded operators on . \ Such relations naturally arise in the context of the operator-valued moment problem, particularly in the study of flat extensions of block Hankel operators. \ Our first goal is to derive an explicit combinatorial formula for . As a concrete application, we provide an explicit expression for the powers of an operator-valued companion matrix. \ In the special case of scalar coefficients , with , we recover a Binet-type formula that allows the explicit computation of the powers and the exponential of algebraic operators in terms of Bell polynomials.

Paper Structure

This paper contains 17 sections, 12 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Our approach
  3. Main contributions
  4. Preliminaries
  5. Notation and terminology
  6. Some known results
  7. An Operator-valued Generalization of a Combinatorial Formula
  8. Computation of powers of the operator-valued companion matrix
  9. The Case of Scalar Coefficients
  10. Algebraic operators
  11. Powers of algebraic operators
  12. Exponentials of algebraic operators
  13. Continuous-Time Operator Recurrence
  14. Declarations
  15. Funding
  16. ...and 2 more sections

Key Result

Theorem 2.1

shmu There exists a unique spectral measure $E$ defined on the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^n)$ such that, for each $k = 1, \dots, n$, where $\sigma(\mathbf{A}) \subseteq \mathbb{R}^n$ is the joint spectrum of the $n$-tuple $\mathbf{A}$, i.e., the support of the spectral measure $E$. Moreover, this joint spectrum satisfies $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (24)

  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Remark 3.1
  • Lemma 3.2
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 14 more