Let be a Banach space, , and . A linear operator generates a C0-semigroup with exactly when is closed, is dense, every real belongs to the resolvent set, andThe same constant must work for every power. When , the first-power bound implies all the power bounds by submultiplicativity. For general , multiplying first-power bounds gives , which is insufficient.[ref]
https://deleo.website/FunctionalAnalysis/section-57.html. Theorem9.7.1; that source uses the opposite sign convention for the resolvent
Necessity follows from the Laplace-transform formula for a semigroup resolvent:The Bochner integral is bounded by . Sufficiency constructs the semigroup from bounded Yosida approximations, using all the resolvent-power estimates for a common growth bound. For and , this characterizes generators of contraction semigroups, providing a route from an unbounded differential linear operator to a well-posed time evolution.
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 .
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The Hille–Yosida theorem is a fundamental result in functional analysis that characterizes the generators of strongly continuous semigroups of linear operators on Banach spaces. It provides a set of conditions under which a certain type of linear operator can be considered the generator of a strongly continuous semigroup. This theorem is particularly important in the study of evolution equations and the analysis of time-dependent systems.