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 (λn)n∈N, we use results obtained in [4] and [5] to give a formula for the ratio λn+1/λn 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.