It is well known that the index is a continuous function on the set of Semi-Fredholm operators on a Banach space, and even on quasi-Banach spaces.
The result is unfortunately false in general Fréchet spaces, as exemplified below.
I am interested in particular in the following problem: proving that, given a continuous function $h\mapsto T_h$ from an interval to the space of semi-fredholm operators, $\text{ind}(T_h)$ is constant. This is, even for Fréchet spaces, the case if we assume $T_h-T_{h'}$ to be compact for $|h'-h|<\varepsilon(h)$: is there any larger class of operators that ensures the same thing? In other words, what is the largest class of operators on a general Frèchet space such that given any semifredholm operator $T$, $T+K$ is Semi-fredholm and has the same index?
Example: $T_h(f):=hxf'+f$, where $T_h$ acts on the space $C^\infty([1,\infty))$. It is easy to see that $T_h$ is surjective for any $h$ and it is not injective for any $h\neq 0$ (indeed its kernel has dimension $1$ for $h\neq 0$). Hence $\text{ind}(T_h)=1_{h\neq 0}$.
