This paper develops some
properties of matrices which have nonnegative elements and act as bounded operators
on one of the sequence spaces lp or lp(μ), where μ is a measure on the integers. Its
chief aim is to relate the operator properties of such matrices to the matrix properties
(the value of the convergence parameter R, R-recurrence and R-positivity) described
in detail in Part I. The relationships between the convergence norm (equal to 1∕R),
the spectral radius, and the operator norm are discussed, and conditions are
set up for the convergence norm to lie in the point spectrum. It is shown
that the correspondence between matrix and operator properties depends
very much on the choice of the underlying space. Other topics considered
are the problem of characterizing the operator norm in matrix terms, and
a general theorem on the structure of a positive contraction operator on
lp.
