Let $T:X\to X$ be a linear operator on a Banach space $X$. 
We know that the spectrum of $T$ is an upper semicontinuous 
function of $T$ for the uniform convergence: that is, if $T_n:X\to X$ is a sequence 
of operators, s.t, $T_n\to T$ uniformly then, given an open set $G$ containing
the spectrum of $T$, $\sigma(T)$, there must exist $k$ s.t  $n>k$  implies  that $\sigma (T_n)$ is contained in $G$.


I would like to know if this result remains valid under 
weaker conditions, like $T_n$ converges to $T$ in
the compact parts of $X$.   

I'd like to know if properties like spectral gap in 
the spectrum remain if we  weaken the notion of  
convergence.