HilleYosida

Hille-Yosida is a fundamental result in functional analysis that characterizes the generators of strongly continuous one-parameter semigroups of linear operators on Banach spaces. It provides precise conditions under which a linear operator governs the time evolution described by a C0-semigroup, in particular contraction semigroups that model dissipative systems.

The theorem (often stated for a densely defined, closed operator A on a Banach space X) states that A generates a C0-semigroup {T(t)} with ||T(t)|| ≤ M e^{ω t} for some M ≥ 1 and ω ∈ R if and only if A is dissipative and, for every λ > ω, the operator λI − A is invertible on X with the resolvent bound ||(λI − A)^{-1}|| ≤ M/(λ − ω). Equivalently, A is m-dissipative (dissipative and maximal in the sense that no proper extension remains dissipative). In the special case M = 1 and ω = 0, this yields a contraction semigroup, with ||(λI − A)^{-1}|| ≤ 1/λ for all λ > 0.

Consequences of the theorem include the well-posedness of linear evolution equations u′(t) = Au(t) with given initial data u(0) = u0, through the semigroup T(t) that provides the mild solution u(t) = T(t)u0. The result is central to the analysis of partial differential equations and other time-dependent problems, linking operator theory to semigroup dynamics.

Historically, the theorem is named after Einar Hille and Kosaku Yosida, who established it in the 1940s. It laid the groundwork for subsequent developments in semigroup theory and its applications, including the Lumer–Phillips theorem for Hilbert spaces.