Let X be the space of complex polynomials on the unit interval [0,1] regarded as a subspace of the complex Banach space C([0,1]) of complex valued functions on [0,1] equipped with the supremum norm. For j=0,1,2,... let f^j be the element of X defined, for t in [0,1] by
Define the mapping T from X to itself, for j= 0,1,2 by Tf^j= (1/j+1)f^j, and elsewhere by linearity. By expressing T as an integral, or otherwise, prove that T is in B(X), T is of norm 1, T maps X one to one onto X, but that the algebraic inverse of T in not in B(X) (where B(X) is the set of bounded linear operators from X into itself).
I can't get started with this question, because I don't know what integral they're looking for. How should T be expressed?