Norm Attaining Operators and Pseudospectrum

It is shown that if 1 < p < infinity and X is a subspace or a quotient of an l(p)-direct sum of finite dimensional Banach spaces, then for any compact operator T on X such that parallel to I + Tk parallel to > 1, the operator I + T attains its norm. A reflexive Banach space X and a bounded rank one operator T on X are constructed such that parallel to I + Tk parallel to > 1 and I + T does not attain its norm.