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A one-parameter group (of unitary operators in a Hilbert space) is a homomorphism of groups
R→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
R→B(X),where B(X) denotes the semigroup of bounded operators X→X.
Typically, we also require a continuity condition such as continuity in the strong topology.
Strongly continuous one-parameter unitary groups (Ut)t≥0 of operators in a Hilbert space H are in bijection with self-adjoint unbounded operators A on H
The bijection sends
A↦(t↦exp(itA)).The operator A is bounded if and only if U is norm-continuous.
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:X→X with dense domain such that any λ>ω belongs to the resolvent set of A and for any λ>ω we have
‖(λI−A)−n‖≤M(λ−ω)−n.[…]
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