By Lindstrum A.
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Next, using the identity R(λ, Du )u = ∞ e−(λ)ξ S(ξ)udξ, λ=0 0 we get the claim. 37) may not hold. We now study properties of uniform spectra of functions in BC(R, X). 1. Let g, f, fn ∈ BC(R, X) such that fn → f as n → ∞ and let Λ ⊂ R be a closed subset. Then the following assertions hold: (i) (ii) (iii) (iv) (v) (vi) spu (f ) = spu (f (h + ·)); spu (αf (·)) ⊂ spu (f ), α ∈ C; sp(f ) ⊂ spu (f ); spu (Bf (·)) ⊂ spu (f ), B ∈ L(X); spu (f + g) ⊂ spu (f ) ∪ spu (g); spu (f ) ⊂ Λ. Proof. (i) - (v) are obvious from the definitions of spectrum and uniform spectrum.
5. Because of pointwise convergence the function g is measurable but not necessarily continous. 6. It is also clear from the definition above that constant functions and continuous almost periodic functions are almost automorphic. 41) is uniform on any compact subset K ⊂ R, we say that f is compact almost automorphic. 19. Assume that f , f1 , and f2 are almost automorphic functions taking values in a Banach space X, φ is a scalar almost automorphic function, and λ is any scalar, then the following hold true.
49) The functions g and h are called respectively the principal and the corrective terms of f . 25. If f is asymptotically almost automorphic then its principal and corrective terms are uniquely determined. Proof. 4 in [N’Gu´er´ekata (79)]. Exercise 11. Prove that every asymptotically almost function is bounded over R+ . April 22, 2008 10:13 World Scientific Book - 9in x 6in Preliminaries stability 37 Exercise 12. Let f ∈ C(R+ , X) and ν ∈ C(R+ , C) be asymptotically almost automorphic. Show that fτ (t) := f (t + τ ), for a fixed τ ∈ R+ and (νf )(t) = ν(t)f (t) are also asymptotically almost automorphic.
Abstract algebra by Lindstrum A.