On the spectral radius of positive operators on Banach sequence spaces

Abstract

Let K 1 , … , K n be (infinite) non-negative matrices that define operators on a Banach sequence space. Given a function f : [ 0 , ∞ ) × … × [ 0 , ∞ ) → [ 0 , ∞ ) of n variables, we define a non-negative matrix f ˆ ( K 1 , … , K n ) and consider the inequality r ( f ˆ ( K 1 , … , K n ) ) ⩽ 1 n r ( K 1 ) + ⋯ + r ( K n ) , where r denotes the spectral radius. We find the largest function f for which this inequality holds for all K 1 , … , K n . We also obtain an infinite-dimensional extension of the result of Cohen asserting that the spectral radius is a convex function of the diagonal entries of a non-negative matrix.