A Combinatorial Formula for Recursive Operator Sequences and Applications

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.