Let $X$ be a Banach space over ${\Bbb C}$, and $K\in K(X)$ ($K(X) = $ compact operators space). Show that if ${\cal L}$ is a maximal chain in the $Lat K$ ($Lat K = $ the collection of all invariant subspaces for $K$), then ${\cal L}$ is a maximal chain in the lattice of all subspaces of $X$.
I do not have any idea about it. Please give me a hint. Thanks in advance.
Let $L$ not be a maximal chain in the lattice of all subspaces of $X$.
$M \notin $ Lat K for some M subspace of X, i.e. $Kx \notin M$ for some $x \in M$.
Suppose that $\{x_n\}$ is a sequence in $M$ such that tends to $x$ in $M$ weakly. By Prop. VI.3.3 [A course in Functional Analysis, John B. Conway] $K$ is completely cont., so that $\{ K{x_n}\}$ converges in norm to $Kx \in M$; this contradicts that $Kx \in M$.