Exercise 5, § 3

Let E and F be infinite-dimensional Banach spaces. A continuous linear mapping $u: E\rightarrow F$ is said to be strictly singular if there does not exist a closed subspace $F_1\subset E$ of infinite dimension such that $u$ defines by passing to the subspaces an isomorphism of $F_1$ onto $u(F_1)$.

a) Let $u$ be a continuous linear mapping of E into F. The following conditions are equivalent:

(i) The mapping $u$ is strictly singular ;

(ii) For every closed subspace $F_1$ of E of infinite dimension, there does not exist a real number $\delta >0$ such that $\|u(x)\|\geqslant \delta \|x\|$ for every $x\in F_1$.

b) Let $u: E\rightarrow F$ be a continuous linear mapping such that the image of $u$ is closed and the kernel of $u$ is of finite dimension. If $v: E\rightarrow F$ is strictly singular then the image of $u+v$ is closed and its kernel is of finite dimension.

c) The set of strictly singular endomorphisms of E is a closed two-sided ideal of $\mathscr{L}(E)$.

d) Every compact linear mapping is strictly singular. If E and F are Hilbert spaces, then every strictly singular mapping $E\rightarrow F$ is compact.

e) There exist Banach spaces E and F and a non-compact strictly singular mapping $E\rightarrow F$.

f) Let $p$ and $r$ be real numbers such that $1\leqslant p < r$. Every continuous linear mapping $\ell^r_K(\mathbf{N})\rightarrow \ell^p_K(\mathbf{N})$ is strictly singular. (Use Exercise 10 of III, p. 106.)