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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTime6 days ago

    Created:

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

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

    RB(X),\mathbf{R} \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

    The bijection sends

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

    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

    […]

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTime6 days ago

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

    diff, v2, current

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