Norm attaining operators and pseudospectrum

Abstract

It is shown that if $1<p<\infty$ and $X$ is a subspace or a quotient of an ${\ell }_p$-direct sum of finite dimensional Banach spaces, then for any compact operator $T$ on $X$ such that $\Vert I+T\Vert >1$, 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 $\Vert I+T\Vert >1$ and $I+T$ does not attain its norm.