Suppose we are given a bounded linear operator $A\colon X\to X$ on a Banach space which is injective and has closed range. Can we find two other operators $T$ and $S$ say such that
$$W=\left[\begin{array}{cc} A & T\\ S & A \end{array} \right] $$
regarded as an operator $W\colon X\oplus X\to X\oplus X$ is
- injective,
- has closed range,
- $\sigma(W)\subset \{a+bi\colon a,b\in \mathbb{R}, b\geqslant 0\}$?
Certainly we can do it when the spectrum of $T$ is finite but I don't know whether this can be generalised further.