Ergodic properties of nonnegative matrices-II

Abstract

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 operatorproperties of such matrices to the matrix properties (the value of the convergence parameter R, UN-recurrence and i?-positivity) described in detail in Part I. The relationships between the convergence norm(equal to 1/22), 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. © 1968 by Pacific Journal of Mathematics.