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Abstract
In this article, we investigate the problem of determining the infinitesimal generator of a co-semigroup on a Hilbert space. The concept of an infinitesimal generator plays a fundamental role in the theory of semigroups and evolution equations, since it characterizes the behavior of the associated operator family. Motivated by the relationship between bounded linear operators and operator groups, we begin by considering a group S(.) generated by a bounded linear operator (A) we have the mapping is differentiable for every (x) in the Hilbert space and satisfies the corresponding differential equation associated This differentiability property provides a natural way to identify an operator that governs the dynamics of the group. Consequently, we obtain a candidate for the infinitesimal generator and investigate the conditions under which this operator serves as the infinitesimal generator of a co-semigroup. The results presented contribute to a better understanding of the connection between co-semigroups and their generators in Hilbert spaces.
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