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Eigenvalues of Totally Positive Integral Operators
Author(s) -
Eveson S. P.
Publication year - 1997
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609396002299
Subject(s) - mathematics , eigenvalues and eigenvectors , spectral radius , operator (biology) , order (exchange) , pure mathematics , mathematics subject classification , combinatorics , simple (philosophy) , matrix (chemical analysis) , kernel (algebra) , biochemistry , physics , chemistry , philosophy , materials science , finance , repressor , quantum mechanics , epistemology , transcription factor , economics , composite material , gene
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.