On the spectral radius of positive operators on Banach sequence spaces

Let K1,…,KnK1,…,Kn be (infinite) non-negative matrices that define operators on a Banach sequence space. Given a function f:[0,∞)×…×[0,∞)→[0,∞)f:[0,∞)×…×[0,∞)→[0,∞) of n   variables, we define a non-negative matrix fˆ(K1,…,Kn) and consider the inequalityr(fˆ(K1,…,Kn))⩽1nr(K1)+⋯+r(Kn),where r denotes the spectral radius. We find the largest function f   for which this inequality holds for all K1,…,KnK1,…,Kn. 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.