Let $H$ be a separable, infinite dimensional, complex Hilbert space. In the book:
Jiang, C. L.; Wang, Z. Y. (1998). Strongly Irreducible Operators on Hilbert Space. CRC press
above the statement of Theorem 5.17 the authors say that given $T \in B(H)$ with disconected spectrum, there exists $\delta >0$ such that $\sigma(T+B)$ is disconnected for all $B$ with $\| B \| < \delta$. If an operator has disconnected spectrum, then it is strongly reducible (i.e. there exists a nontrivial projection commuting with it), while the converse is not necessarily true. Moreover, in the introduction of the paper:
Herrero, D. A.; Jiang, C. L. Limits of strongly irreducible operators, and the Riesz decomposition theorem. Michigan Math. J. 37 (1990), no. 2, 283-291
it is stated that if $T$ has disconnected spectrum, all the operators which are "close enough" to $T$ commute with a nontrivial (Riesz) projection, and hence are strongly reducible.
I think that "close enough" here means 'the perturbation $B$ has a small norm'. In what follows, we will consider these two constants to be the largest possible.
As a first question, I wonder if there is any relation between the two constants in the above paper (I think that they may be equal, but I'm not sure about this).
In any case, is there any known way to evaluate these two constants? Or is there any known upper/lower bound?
