nLab one-parameter semigroup

1. Idea

A one-parameter group (of unitary operators in a Hilbert space) is a homomorphism of groups

$$\mathbf{R} \to U(H),$$

where $H$ is a Hilbert spaces, $U(H)$ denotes its group of unitary operators and $\mathbf{R}$ the additive group of real numbers.

More generally, one can define one-parameter semigroups of operators in a Banach space $X$ as homomomorphisms of monoids

$$\mathbf{R}_{\ge0} \to B(X),$$

where $B(X)$ denotes the semigroup of bounded operators $X\to X$.

Typically, we also require a continuity condition such as continuity in the strong topology.

2. Stone theorem

Strongly continuous one-parameter unitary groups $(U_t)_{t\ge0}$ of operators in a Hilbert space $H$ are in bijection with self-adjoint unbounded operators $A$ on $H$:

This bijection sends

$$A \mapsto \big( t\mapsto \exp(itA) \big) \,.$$

The operator $A$ is bounded if and only if $U$ is norm-continuous.

3. Hille–Yosida theorem

Strongly continuous one-parameter semigroups $T$ of bounded operators on a Banach space $X$ (alias $C_0$-semigroups) satisfying $\|T(t)\|\le M\exp(\omega t)$ are in bijection with closed operators $A\colon X\to X$ with dense domain such that any $\lambda\gt \omega$ belongs to the resolvent set of $A$ and for any $\lambda\gt\omega$ we have

$$\|(\lambda I-A)^{-n}\|\le M (\lambda-\omega)^{-n}.$$

4. Related concepts

• exponential

5. References

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See also

• Wikipedia, $C_0$ semigroup