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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 20th 2022
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: 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]] and $U(H)$ denotes its group of [[unitary operators]]. More generally, one can define __one-parameter semigroups__ of operators in a [[Banach space]] $X$ as homomomorphisms of groups $$\mathbf{R} \to B(X),$$ 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]]. ## 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 $H$ The bijection sends $$A\mapsto (t\mapsto \exp(itA)).$$ The operator $A$ is bounded if and only if $U$ is norm-continuous. ## 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 $$\|(\lambda I-A)^{-n}\|\le M (\lambda-\omega)^{-n}.$$ ## References […] <a href="https://ncatlab.org/nlab/revision/one-parameter+semigroup/1">v1</a>, <a href="https://ncatlab.org/nlab/show/one-parameter+semigroup">current</a>

   Created:

   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 and $U(H)$ denotes its group of unitary operators.

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

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

   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.

   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 $H$

   The bijection sends

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

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

   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

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

   References

   […]

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 20th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added cross-link with the entry *[[U(ℋ)]]*. <a href="https://ncatlab.org/nlab/revision/diff/one-parameter+semigroup/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/one-parameter+semigroup/2">v2</a>, <a href="https://ncatlab.org/nlab/show/one-parameter+semigroup">current</a>

   I have added cross-link with the entry U(ℋ).

   diff, v2, current