Semigroups of Operators

Semigroups of operators are mathematical tools used to analyze how systems evolve over time, especially in contexts like differential equations and dynamical systems. Think of them as a family of transformations, each representing the system's state after a certain time interval. These transformations follow specific rules: applying two sequential operators is the same as applying one that covers the combined time, and starting from an initial state, the operators describe its progression smoothly. Semigroups provide a structured way to study continuous change, helping mathematicians and scientists understand how complex processes develop step-by-step.