Fredholm Properties of Evolution Semigroups
نویسنده
چکیده
We show that the Fredholm spectrum of an evolution semigroup {E}t≥0 is equal to its spectrum, and prove that the ranges of the operator Et − I and the generator G of the evolution semigroup are closed simultaneously. The evolution semigroup is acting on spaces of functions with values in a Banach space, and is induced by an evolution family that could be the propagator for a well-posed linear differential equation u′(t) = A(t)u(t) with, generally, unbounded operators A(t); in this case G is the closure of the operator G given by (Gu)(t) = −u′(t) +A(t)u(t).
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