StronglyContinuousOneParameterSemigroup - crowlogic/arb4j GitHub Wiki

A strongly continuous one-parameter semigroup, often found in the context of functional analysis and the study of partial differential equations, is a family of linear operators that generalizes the concept of exponential functions to operator theory. Here are its key features:

1. Family of Linear Operators: It is a family ${T(t)}$ of linear operators on a Banach space $X$, where $t \geq 0$.

2. Semigroup Property: It satisfies the semigroup property: $T(0)$ is the identity operator on $X$ and

$$T(t + s) = T(t)T(s) \forall t, s \geq 0$$

3. Strong Continuity: The semigroup is "strongly continuous", meaning that $T(t)x$ is continuous in $t$ for each fixed $x$ in $X$.

$$\lim_{t \rightarrow 0} T(t)x = x \forall x \in X$$

These semigroups are particularly important in solving linear partial differential equations, where they describe the evolution of the system over time. The operator $T(t)$ can be thought of as advancing the state of the system by time $t$. The concept of strong continuity ensures that small changes in time lead to small changes in the state of the system, a property often desired in physical systems.