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 .
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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.