Almost automorphic groups and semigroups in Fréchet spaces (Q2455782)

The authors study the space \(AA(\mathbb{R},X)\) of almost automorphic functions defined on \(\mathbb{R}\) with values in a Fréchet space \(X\). Similar to the case of Banach spaces, \(AA(\mathbb{R},X)\) is a Fréchet space. Semigroups of operators on Fréchet spaces are recalled in Section 3. In Section 4, a uniformly bounded one parameter group \(T(t)\) of bounded linear operators on \(X\) is defined. It is shown that \(T(t)x_0\) is (asymptotically) almost automorphic if and only if \(T(t)x_0\) is (asymptotically) almost periodic. The proofs are similar to the case when \(X\) is a Banach space.