Let $X$, $Y$ be Banach spaces. The set of all bounded linear operators from $X$ to $Y$ is denoted by $L(X,Y)$. $L(X,Y)$ becomes a Banach space with the operator norm.


The set of all compact linear operators from $X$ to $Y$ is denoted by $K(X,Y)$. 


It is well known that $K(X,Y)$ is a closed subspace of $L(X,Y)$.


**My question**


Assume $\{T\_n\}\_{n=1}^{\infty} \subset K(X,Y)$ strongly converges to some $T \in L(X,Y)$. That is, for all $x \in X$, $T\_{n}x \to Tx$ in $Y$.


Is there any sufficient conditions such that $T\_{n}$ converges to $T$ with respect to operator norm? 



Additional conditions for uniform convergence
Hard to know what you are looking for but a necessary and sufficient cndition would be that the sequence be relatively compact in the norm topology.


