On the submultiplicativity of norms of Hölder and Minkowski type

Let ν1and ν2 be absolute norms in . The submultiplicativity of the norms H12 and M12 in is studied. In general, neither the condition “the norm μ has a lub minorant” nor "the norm μ is spectrally dominant " is equivalent to the condition "μ is submultiplicative". The equivalence is proved in the case μ=M12. Finally, it is seen that in testing these conditions it is sufficient to restrict considerations to matrices of rank 1.