Functional Analysis Answers

state and prove spectral mapping theorem for polynomial

T η(x) = { ηi.xi} from i=1 to infinity Show that T η is bounded, determine its spectrum and spectral radius. When is the point spectrum of Tf non-empty.

Let TεB(X,X) be invertible in B(X , X) , X is a complex Banach space.Show that σ(T^ -1)={1/λ : λЄσ(T)}

Mikel,drogba and wayne were known to have tenty,ten and twenty out of fifty fired penalty kicks out of which a number of scores were recorded in the last league. If the operation in the scores were recorded to imply that mikel had six scores more than drogba in 2011,mikel obi scored square as much as the score of drogba while wayne had no score through out.
1: present this information explicitly
2: determine wether the information is a linear map or not.
Its a functional analysis problem

Let E be a dense linear subspace of a normed vector space X, and let Y be a
Banach space. Suppose T_0 in £(E, Y) is a bounded linear operator from E to Y.
Show that To can be extended to T in £(X, Y) (by continuity) without increasing
its norm.