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Strongly Continuous Semigroup

A strongly continuous semigroup, or C_0-semigroup, on a Banach space X is a semigroup (T(t))_(t>=0) of bounded linear operators satisfying

T(0)=I,
(1)
T(t+s)=T(t)T(s),
(2)

for s,t>=0, and the strong continuity condition

lim_(t->0+)T(t)x=x,
(3)

for every x in X. Here I is the identity operator. Its infinitesimal generator is the generally unbounded linear operator A defined by

Ax=lim_(t->0+)(T(t)x-x)/t
(4)

on the domain consisting of the x for which this limit exists. Every strongly continuous semigroup satisfies an exponential growth bound

||T(t)||<=Me^(omegat)
(5)

for some M>=1 and real omega. Such semigroups provide the operator-theoretic formulation of autonomous linear evolution equations.

See also

Contraction Semigroup, Dissipative Operator, Infinitesimal Generator, Lumer-Phillips Theorem, Semigroup, Semigroup Stability, Strong Stability

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References

Engel, K.-J. and Nagel, R. One-Parameter Semigroups for Linear Evolution Equations. New York: Springer-Verlag, 2000.Pazy, A. Semigroups of Linear Operators and Applications to Partial Differential Equations. New York: Springer-Verlag, 1983.

Cite this as:

Weisstein, Eric W. "Strongly Continuous Semigroup." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/StronglyContinuousSemigroup.html

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