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 \(\mathcal{H}\) that satisfy a linear recurrence relation of the form $$T_{n+r}=A_0{T}_n+A_1{T}_{n+1}+\cdots +A_{r-1}{T}_{n+r-1}(\text{for all }n\ge 0),$$ 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 $A_k=a_k{I}_{\mathcal{H}}$, 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.