Bounded linear operators and bounded invertibles

Question

Problem:

Bounded Linear Operators and Bounded Invertibles

Let ε = c[0,1] = {ƒ : [0,1] →C | ƒ is a continuous function}.

Let ||ƒ||_∞ = sup {| ƒ(t)|: t∈[0,1]} ,:for each f in ε Define T: ε →by

                            (T(ƒ))(t) = ∫^t₀ ƒ(s)ds

     for each t ∈[0,1], and for each f in ε.

a) Show that is a bounded linear operator on ε .

b) Compute ||Tⁿ||, For each n in N, and compute σ(T) .

c) Suppose that g . Show that the integral equation

               ƒ(t) - ∫^t₀ƒ(s)ds = g(t) for each t in [0,1]