Norms and lower bounds of operators on the Lorentz sequence space $d(w,1)$

Abstract

Conditions are found under which the norm of an operator on a Banach sequence space is determined by its action on decreasing, positive sequences. For the space $d(w,1)$, the norm and “lower bound” of such operators can be equated to the supremum and infimum of a certain sequence. These quantities are evaluated for the averaging, Copson and Hilbert operators, with the weighting sequence given either by $w = 1/n^{\alpha}$ or by the corresponding integral.