nLab one-parameter semigroup

Contents

Context

Group Theory

Operator algebra

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Contents

Idea

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

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

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

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

R 0B(X),\mathbf{R}_{\ge0} \to B(X),

where B(X)B(X) denotes the semigroup of bounded operators XXX\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) t0(U_t)_{t\ge0} of operators in a Hilbert space HH are in bijection with self-adjoint unbounded operators AA on HH:

This bijection sends

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

The operator AA is bounded if and only if UU is norm-continuous.

Hille–Yosida theorem

Strongly continuous one-parameter semigroups TT of bounded operators on a Banach space XX (alias C 0C_0-semigroups) satisfying T(t)Mexp(ωt)\|T(t)\|\le M\exp(\omega t) are in bijection with closed operators A:XXA\colon X\to X with dense domain such that any λ>ω\lambda\gt \omega belongs to the resolvent set of AA and for any λ>ω\lambda\gt\omega we have

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

References

[…]

See also

Last revised on June 21, 2022 at 03:35:23. See the history of this page for a list of all contributions to it.