# strongly continuous semigroup

## strongly continuous semigroup

[¦strȯŋ·lē kən¦tin·yə·wəs ′sem·i‚grüp]
(mathematics)
A semigroup of bounded linear operators on a Banach space B, together with a bijective mapping T from the positive real numbers onto the semigroup, such that T (0) is the identity operator on B, T (s + t) = T (s) T (t) for any two positive numbers s and t and, for each element x of B, T (t) x is a continuous function of t.
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References in periodicals archive ?
where A : D(A) [subset or equal to] X [right arrow] X is the infinitesimal generator of strongly continuous semigroup of bounded linear operators T(t), t > 0.
[{[B.sub.t]}.sub.t[greater than or equal to]0] is a strongly continuous semigroup. Proof.
One can verify that the family [P.sup.w] = [([P.sup.w.sub.t]).sub.t[greater than or equal to] 0] forms a strongly continuous semigroup of contraction operators on [L.sup.2]([OMEGA]).
Denote by [DELTA] = [[summation].sup.m.sub.j=1]([[partial derivative].sup.2]/[partial derivative][x.sup.2.sub.j]) the Laplace operator, with domain D([DELTA]) = [W.sup.1,2.sub.0](Y) [intersection] [W.sup.2,2.sub.0](Y), which generates a strongly continuous semigroup [mathematical expression not reproducible], where [W.sup.1,2.sub.0](Y) and [W.sup.2,2.sub.0](Y) are the Sobolev spaces with compactly supported sets.
where x(*) the state of the system (1.1) takes values in a Banach space X (state space), the unbounded operator (A,D(A)) generates a [C.sub.0]-semigroup (strongly continuous semigroup) [(T(t)).sub.t[greater than or equal to]0] on X, B is unbounded linear control operators in the sense that it is a bounded linear operator from X to a larger Banach space V [contains] X, i.e.
Motivated by the works of Mustari and Taylor [15,17], we have recently begun to study the mean ergodic theorem under the framework of RN modules  to obtain the mean ergodic theorem in the sense of convergence in probability, where we proved the mean ergodic theorem for a strongly continuous semigroup of random unitary operators defined on complete random inner product modules (briefly, complete RIP modules).
Then A generates a strongly continuous semigroup that is analytic, and resolvent operator R(t) can be extracted from this analytic semigroup(see ).
where A:D(A) [subset] X [right arrow] X is the infinitesimal generator of a strongly continuous semigroup [{T(t)}.sub.t[greater than or equal to]0], and B: X [right arrow] [??](H, X) is a bounded linear operator.
i) The two-parameter (nonlinear) semiflow [sigma] is continuous if and only if it induces a strongly continuous semigroup [{T(t)}.sub.t[greater than or equal to]0] on X by the formula:
where A generates a strongly continuous semigroup ([e.sup.tA])t [greater than or equal to] 0 on an infinite dimensional Banach space Y (state space) whose norm will be denoted by [parallel].[parallel], B : D(B)([subset] Y) [arrow right] Y is a possibly unbounded linear operator and the control function u(*) denotes the scalar control.
After a short introduction to the theory of strongly continuous semigroups on linear operators on Banach spaces, the authors introduce abstract delay equations on a Banach space X and discuss examples.
van Casteren, Generators of Strongly Continuous Semigroups, Research Notes in Math., Pitman Publishing Limited, London, 1985.

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