Lumer-Phillips Theorem

The Lumer-Phillips Theorem provides criteria to determine when a linear operator generates a semigroup of transformations that describe how systems evolve over time, especially in contexts like differential equations. Essentially, it states that if an operator is densely defined, dissipative (it doesn't increase the system's energy), and its range shifted by a small amount covers the whole space, then it guarantees the existence of a well-behaved evolution process (a semigroup). This theorem helps mathematicians verify when certain equations have solutions that change smoothly over time, ensuring stable and predictable system behavior.