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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 20th 2022

    Created:

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

    RU(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

    RB(X),

    where B(X) denotes the semigroup of bounded operators XX.

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

    Stone theorem

    Strongly continuous one-parameter unitary groups (Ut)t0 of operators in a Hilbert space H are in bijection with self-adjoint unbounded operators A on H

    The bijection sends

    A(texp(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 C0-semigroups) satisfying T(t)Mexp(ωt) are in bijection with closed operators A:XX with dense domain such that any λ>ω belongs to the resolvent set of A and for any λ>ω we have

    (λIA)nM(λω)n.

    References

    […]

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 20th 2022

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

    diff, v2, current