Question

if T belongs to b(X,Y) is not compact can the restriction of T to an infinite dimensional subspace of X be compact?

Answer 1

Yes. Look at the following example. Take any infinite dimensional Banach spaces X and Y. Let S∈K(X), then restriction of T:=S⊕∞1Y∈B(X⊕∞Y) to X is S which is compact by construction.