TY  - JOUR
T1  - A combinatorial formula for recursive operator sequences and applications
AU  - Curto, Raul
AU  - Ech-Charyfy, Abderrazzak
AU  - Idrissi, Kaissar
AU  - Zerouali, El Hassan
PY  - 2025
DA  - December
Y2  - 2025
DO  - 10.33205/cma.1809730
JF  - Constructive Mathematical Analysis
JO  - CMA
PB  - Tuncer ACAR
WT  - DergiPark
SN  - 2651-2939
SP  - 200
EP  - 216
VL  - 8
IS  - 4
LA  - en
AB  - 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.
KW  - Operator recurrence relation
KW  - operator-valued moment problem
KW  - Binet formula
KW  - combinatorial formula
KW  - companion matrix
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UR  - https://doi.org/10.33205/cma.1809730
L1  - https://dergipark.org.tr/en/download/article-file/5357521
ER  -