Hilbert evolution algebras, weighted digraphs, and nilpotency (Q6577483)

Hilbert evolution algebras generalize the concept of finite-dimensional evolution algebras by incorporating the structure of Hilbert spaces and with dimension infinity. Let \(A\) be a real or complex separable Hilbert space which is equipped with an algebra structure. Then \(A\) is a separable Hilbert space if there exists an orthonoral basis \(\{e_i\}_{i\in \mathbb{N}}\), such that \(e_i\cdot e_j =0\) for all \(i\ne j\), and for each \(a\in A\), the left multiplication \(L_a\) is a continuous function.\N\NSection 2 establishes basic notations and definitions for infinite weighted digraphs. In Section 3, the authors introduce the weighted digraph associated with a given Hilbert evolution algebra and give related properties and examples. Section 4 extends the discussion of nilpotency in evolution algebras from [\textit{A. Elduque} and \textit{A. Labra}, Linear Multilinear Algebra 69, No. 2, 331--342 (2021; Zbl 1481.17005)] to the context of Hilbert evolution algebras.