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Infinitesimal Generator

The infinitesimal generator of a strongly continuous semigroup (T(t))_(t>=0) on a Banach space X is the linear operator A defined by

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

for every x for which the limit exists. These vectors form the domain of A. The infinitesimal generator is a densely defined closed operator and uniquely determines the semigroup. For a contraction semigroup, the Lumer-Phillips theorem characterizes the infinitesimal generator as an m-dissipative operator.

See also

Contraction Semigroup, Dissipative Operator, Lumer-Phillips Theorem, Strongly Continuous Semigroup

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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. "Infinitesimal Generator." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/InfinitesimalGenerator.html

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