Eigenvalues of totally positive integral operators

S P Eveson

doi:10.1112/S0024609396002299

Eigenvalues of totally positive integral operators

S P Eveson

Mathematics

Research output: Contribution to journal › Article › peer-review

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
Pages (from-to)	216-222
Number of pages	7
Journal	Bulletin of the london mathematical society
Volume	29
Issue number	2
DOIs	https://doi.org/10.1112/S0024609396002299
Publication status	Published - Mar 1997

Keywords

BIRKHOFF-HOPF THEOREM

Access to Document

10.1112/S0024609396002299

http://blms.oxfordjournals.org/cgi/reprint/29/2/216

Cite this

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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 formular(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].",

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AB - 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 formular(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].

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