Hilbert evolution algebras, weighted digraphs, and nilpotency

Hilbert evolution algebras generalize evolution algebras through a framework of Hilbert spaces. In this work we focus on infinite-dimensional Hilbert evolution algebras and their representation through a suitably defined weighted digraph. By means of studying such a digraph we obtain new properties for these structures extending well-known results related to the nilpotency of finite dimensional evolution algebras. We show that differently from what happens for the finite dimensional evolution algebras, the notions of nil and nilpotency are not equivalent for Hilbert evolution algebras. Furthermore, we exhibit necessary and sufficient conditions under which a given Hilbert evolution algebra is nil or nilpotent. Our approach includes illustrative examples.