Preprint
A Combinatorial Formula for Recursive Operator Sequences and Applications
ArXiv.org
Cornell University
04/05/2026
DOI: 10.48550/arxiv.2604.04320
Abstract
We study sequences of bounded operators (Tn)n≥0 on a complex separable Hilbert space H that satisfy a linear recurrence relation of the form Tn+r = A0Tn + A1Tn+1 + · · · + Ar−1Tn+r−1 (for all n ≥ 0), where the coefficients A0, A1, . . . , Ar−1 are pairwise commuting bounded operators on 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 Tn. 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 Ak = ak IH, with ak ∈ 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.
Details
- Title: Subtitle
- A Combinatorial Formula for Recursive Operator Sequences and Applications
- Creators
- Raul E CurtoAbderrazzak Ech-charyfyKaissar IdrissiEl Hassan Zerouali
- Resource Type
- Preprint
- Publication Details
- ArXiv.org
- DOI
- 10.48550/arxiv.2604.04320
- ISSN
- 2331-8422
- Publisher
- Cornell University; Ithaca, New York
- Language
- English
- Date posted
- 04/05/2026
- Academic Unit
- Mathematics
- Record Identifier
- 9985151591302771
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