HilleYosidaTheorem - crowlogic/arb4j GitHub Wiki

Hille–Yosida theorem: Let $X$ be a Banach space. A linear operator $A : D(A) \subseteq X \rightarrow X$ is the infinitesimal generator of a $C_0$ semigroup if and only if the following conditions are satisfied:

  1. Dense domain: The domain $D(A)$ of $A$ is dense in $X$. (That is, for every element $x$ in $X$, there exists a sequence in $D(A)$ that converges to $x$).

  2. Closed graph: The graph of $A$ is closed in $X \times X$. (A graph is a set of ordered pairs where the first element comes from $D(A)$ and the second element is the image of the first under $A$. Saying the graph is closed means that if you have a sequence of points in the graph that converges to some point in $X \times X$, then this limit point is also in the graph).

  3. Resolvent condition: For every complex number $\lambda$ with $\text{Re}(\lambda) > \omega$ for some $\omega$ in $\mathbb{R}$, the operator $\lambda I - A$ has a bounded inverse that belongs to the space of bounded linear operators on $X$ ($I$ is the identity operator, and $\text{Re}(\lambda)$ denotes the real part of $\lambda$). This inverse is called the resolvent of $A$ at $\lambda$, and its norm is required to be bounded by a function of the form $\frac{M}{\text{Re}(\lambda) - \omega}$ for some $M > 0$.

Each of these conditions is non-trivial. The first one says that the operator can "reach" close to any point in the space. The second one is a regularity condition that ensures that the operator behaves well with respect to limits. The third one is the key condition that links the operator to the exponential function, and it's the trickiest one to check in practice.

This theorem is a central result in the theory of $C_0$ semigroups and has many applications in differential equations and functional analysis. For instance, it allows us to construct solutions to certain differential equations from their generators, and vice versa.