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Question #8296

If S,T Є B(X,X),show that for any λ Є ρ(S) intersection ρ(T),
Rλ(S) - Rλ(T) = Rλ(S) (T – S) Rλ(T)

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**1.**Show that K(x-a) = O(k-a) [this is the letter O no the number 0] for any K.**2.**state and prove spectral mapping theorem for polynomial**3.**T η(x) = { ηi.xi} from i=1 to infinity Show that T η is bounded, determine i**4.**Tη(x) = {ηi.xi} from i=1 to infinity Show that Tη is bounded, determine its**5.**Let TεB(X,X) be invertible in B(X , X) , X is a complex Banach space.Show that &s**6.**A subspace Y of a normed space X is said to be invariant under a linear operator T:X→X if T**7.**Let TεB(X,X) ,X is a comlex Banach space and T be idempotent(i.e T^2=T) .Prove that if

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