A Combinatorial Formula for Recursive Operator Sequences and Applications

TL;DR

This paper derives a combinatorial formula for recursive sequences of bounded operators on Hilbert spaces, with applications to operator powers, exponentials, and the operator-valued moment problem.

Contribution

It provides an explicit combinatorial formula for operator sequences satisfying linear recurrence relations with commuting coefficients, extending classical formulas to operator contexts.

Findings

Explicit formula for operator sequences satisfying recurrence relations.

Application to powers of operator-valued companion matrices.

Recovery of a Binet-type formula for scalar coefficients using Bell polynomials.

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.