  The main result of the paper shows that, for 1<p and 1<=q, a linear operator
T from l_p to l_q attains its norm if, and only if, there exists a not weakly
null maximizing sequence for T (counterexamples can be easily constructed when
p=1). For 1<p (and q different from p), as a consequence of the previous result
we show that any not weakly null maximizing sequence for a norm attaining
operator T from l_p to l_q has a norm-convergent subsequence (and this result
is sharp in the sense that it is not valid if p=q). We also investigate
lineability of the sets of norm-attaining and non-norm attaining operators.
