Eigenvalues of Totally Positive Integral Operators

It is known [ 10 , 11 ] that if T is an integral operator with an extended totally positive kernel, then T has a countably infinite family of simple, positive eigenvalues. We prove a similar result for a rather larger class of kernels and, writing the eigenvalues of T in decreasing order as (λ n ) n ∈N , we use results obtained in [ 4 ] and [ 5 ] to give a formula for the ratio λ n +1 /λ n analogous to that given in [ 3 ] for the case of a strictly totally positive matrix, and to the spectral radius formula r ( T ) = lim n → ∞‖ T n ‖1 / n = inf n ∈ N‖ T n ‖1 / n .This may be regarded as a generalisation of inequalities due to Hopf [ 8 , 9 ]. 1991 1991 Mathematics Subject Classification 47G10, 47B65.