Schauder Bases and Operator Theory II: (SI) Schauder Operators
  In this paper, we will show that for an operator $T$ which is injective and has dense range, there exists an invertible operator $X$ (in fact we can find $U+K$, where $U$ is an unitary operator and $K$ is a compact operator with norm less than a given positive real number) such that $XT$ is strongly irreducible. As its application, strongly irreducible operators always exist in the orbit of Schauder matrices. 
Geng Tian, Youqing Ji, Yang Cao
math.FA