The Hille Yosida theorem provides a necessary and sufficient condition for a closed linear operator " A " on a Banach space to be the infinitesimal generator of a strongly continuous one-parameter semigroup.
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This condition is necessary, as there exist closed linear operators that are unbounded ( not continuous ); a prototypical example is provided by the derivative operator on " C ( [ 0, 1 ] ) " ( whose domain is a strict subset of " C ( [ 0, 1 ] ) " ).
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