## Abstract

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 (lambda(n))(n is an element of N), we use results obtained in [4] and [5] to give a formula for the ratio lambda(n+1)/lambda(n) analogous to that given in II for the case of a strictly totally positive matrix, and to the spectral radius formula

r(T) = lim/n-->infinity parallel to T-n parallel to(1/n) = inf/n is an element of N parallel to T-n parallel to(1/n).

This may be regarded as a generalisation of inequalities due to Hopf [8, 9].

Original language | English |
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Pages (from-to) | 216-222 |

Number of pages | 7 |

Journal | Bulletin of the london mathematical society |

Volume | 29 |

Issue number | 2 |

DOIs | |

Publication status | Published - Mar 1997 |

## Keywords

- BIRKHOFF-HOPF THEOREM