A new method for constructing invariant subspaces
  The method of compatible sequences is introduced in order to produce non-trivial (closed) invariant subspaces of (bounded linear) operators. Also a topological tool is used which is new in the search of invariant subspaces: the extraction of continuous selections of lower semicontinuous set valued functions. The advantage of this method over previously known methods is that if an operator acts on a reflexive Banach space then it has a non-trivial invariant subspace if and only if there exist compatible sequences (their definition refers to a fixed operator). Using compatible sequences a result of Aronszajn-Smith is proved for reflexive Banach spaces. Also it is shown that if $X$ be a reflexive Banach space, $T \in {\mathcal L} (X)$, and $A$ is any closed ball of $X$, then either there exists $v \in A$ such that $Tv=0$, or there exists $v \in A$ such that $\bar{\text{Span}} \text{Orb}_T (Tv)$ is a non-trivial invariant subspace of $T$, or $A \subseteq \bar{\text{Span}} \{T^k x_{\ell} : \ell \in {\mathbb N}, 1 \leq k \leq \ell \} $ for every $(x_n)_n \in A^{\mathbb N}$. 
George Androulakis