C0Semigroup

A C0-semigroup, or strongly continuous one-parameter semigroup, is a family {T(t)}_{t≥0} of bounded linear operators on a Banach space X that satisfies: T(0) = I, T(s+t) = T(s)T(t) for all s,t ≥ 0, and for each x ∈ X, the map t ↦ T(t)x is continuous on [0, ∞). Such semigroups describe time evolution in various linear systems and PDEs.

The infinitesimal generator A of a C0-semigroup is defined by its domain D(A) = {x ∈ X : lim_{t→0+} (T(t)x − x)/t exists} and Ax = lim_{t→0+} (T(t)x − x)/t for x ∈ D(A). The generator is closed and densely defined, and it determines the semigroup through the abstract Cauchy problem u′(t) = Au(t), u(0) = x0, where mild solutions are given by u(t) = T(t)x0 when x0 ∈ X.

Growth and resolvent: The growth bound ω0 = inf{ω ∈ R : ∥T(t)∥ ≤ M e^{ω t} for some M ≥ 1} characterizes the exponential behavior of T(t). If Re λ > ω, the resolvent R(λ,A) = (λI − A)⁻¹ exists and can be represented as R(λ,A) = ∫_0^∞ e^{-λ t} T(t) dt. The Hille–Yosida theorem provides necessary and sufficient conditions for a densely defined closed operator to generate a C0-semigroup with given growth bounds, in terms of resolvent estimates.

Examples include the heat semigroup on L^p(R^n) given by Gaussian convolution, and the shift semigroup (T(t)f)(x) = f(x+t) on L^p(R). Finite-dimensional semigroups arise from bounded generators, with T(t) = e^{tA}.

Applications of C0-semigroups span the abstract formulation of linear evolution equations, population dynamics, quantum mechanics, and control theory, where T(t) describes the time evolution of states and A governs infinitesimal dynamics.