Exercise 19, § 2

Let X be a complex Banach space and $x\in A =\mathscr{L}(X)$.

a) If $x$ is not invertible, then $x$ is a topological left or right zero divisor. (Consider successively the following cases:

$1^{\circ})x$ is non-injective ;

$2^{\circ})\overline{x(X)}\not= X$ ;

$3^{\circ})x$ is injective, $x(X)$ is dense in X and distinct from X.)

b) If $x$ maps bicontinuously X onto a closed vector subspace of X distinct from X, or if $x$ is non-injective and has image X, then $x$ is interior to the set of non-invertible elements of A.