The semigroup property is for nonnegative times, together with . It expresses that evolving for two consecutive intervals is equivalent to evolving for their total duration.
An operator family is strongly continuous when is norm-continuous for every vector . This is continuity in the strong operator topology.
The infinitesimal generator is the generally unbounded operatoron the vectors for which the norm limit exists.
The generator domain consists of the vectors whose semigroup orbit is right-differentiable at zero. It is a dense linear subspace, and the generator is closed.
Equip with its graph norm. The restrictions form a C0-semigroup on this Banach space; its generator is restricted to .
The Hille-Yosida theorem characterizes generators of exponentially bounded C0-semigroups by closedness, dense domain, a resolvent half-line, and uniform bounds on every positive resolvent power.
If generates a C0-semigroup satisfying , then for its resolvent operator isRepeated differentiation gives .
For in the underlying Banach space,belongs to the generator domain, satisfies , and converges to as .
A time-indexed family is stable with constants when products of its frozen semigroups obeyfor all nonnegative . This condition controls products uniformly as the number of time slices grows.
On a Hilbert space, an operator is dissipative when for every . This is the infinitesimal form of norm contraction.
A maximal dissipative operator is dissipative and has no proper dissipative extension. Equivalently, for a densely defined dissipative operator, is onto for some positive .
The Lumer-Phillips theorem says that a densely defined operator generates a contraction C0-semigroup exactly when it is maximal dissipative.
A strongly continuous unitary group is a family for all real that is strongly continuous, obeys the group law, and preserves the Hilbert norm. Its generator is skew-adjoint; conversely, every skew-adjoint operator generates such a group.
An operator is skew-adjoint when . Both and are then maximal dissipative, and the generated group is unitary.
An abstract Cauchy problem has the form in a Banach space, where may be unbounded and generates the homogeneous evolution.
A nonautonomous abstract Cauchy problem has the form , with a time-dependent generally closed linear operator. Under stability, common-domain, and regularity hypotheses, it is propagated by an evolution family.
An evolution family consists of bounded operators for satisfying and . For sufficiently regular data it solves and .
For a partition , the frozen-generator approximation isUnder the hypotheses of the nonautonomous generation theorem, these products converge strongly and uniformly on compact time triangles to the evolution family.
A mild solution need only be continuous and satisfy the integrated semigroup formula. It need not lie in the generator domain or be differentiable.
A semilinear abstract Cauchy problem has , where the linear part generates a C0-semigroup and the nonlinear map is typically locally Lipschitz on the phase space.
A local mild solution is a fixed point of the nonlinear variation-of-constants map in . Local Lipschitz continuity of the nonlinearity gives existence and uniqueness for sufficiently small .
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A \( C_0 \)-semigroup (also known as a strongly continuous semigroup) is a mathematical object used in the context of functional analysis and the theory of linear operators. It is particularly relevant in the study of linear differential equations and partial differential equations, as well as in the analysis of dynamical systems. ### Definition Let \( X \) be a Banach space.