Let $H$ be a complex, infinite-dimensional, separable Hilbert space. Let $T \in B(H)$ be an operator with disconnected spectrum. In the introduction of the paper:
Jiang, C.; Sun, S.; Wang, Z. (1997). Essentially normal operator +compact operator = strongly irreducible operator.Transactions of the American Mathematical Society, vol 349, 1, 217-233
it is stated that $T+K$ is strongly reducible for every compact operator $K$. I know that any operator $A$ having disconnected spectrum cannot be strongly irreducible, but how can we prove the the above sum is never strongly irreducible, whichever is $K$ compact? Any suggestion is greatly appreciated.
EDIT: to be more precise, on the first page of the link in the comments, it is stated that:
... P. R. Halmos proved that for each $T \in \mathcal{L}(H)$ and $\epsilon > 0$, there exists a $K$ compact with $\| K \| < \epsilon$ such that $T+K$ is irreducible [11]. It is obvious that if $\sigma(T)$ is not connected, then for each $K$ compact, $T+K$ is still strongly reducible. Thus D. A. Herrero asked the following questions ...