For a bounded linear
operator T from an Lp to an Lq space (1 ≤ p,q < ∞), we study its norming vectors
i.e. those, including the zero vector, on which T attains its norm. The scalar field
may be the reals or the complex numbers. Our first two main results are the
characterization of the set of norming vectors for a positive T when both p > 1 and
either (i) p = q or (ii) p > q. The descriptions may not hold if T is not positive, but
they do in modified forms if |T| exists with norm ∥T∥. We also prove that if p > q
and one of the two underlying measures is purely atomic, then every regular T is
norm-attaining. Sufficient conditions for T (of norm 1) to be an extreme contraction
in the case p > q > 1 are derived from properties of its norming vectors. All
results extend to the case of quaternion scalars with little change of the
proofs.
