The ideal structure of the algebraic eigenspace to the spectral radius of eventually compact, reducible, positive linear operators
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The classical Perron-Frobenius theory, concerning the distribution of Eigen-values of a nonnegative square matrix A, has been applied in recent years to study the positivity structure of the algebraic Eigen-space belonging to the spectral radius of A. The most complete result is by U.G. Rothblum [Linear Algebra and its Applications 12 (1975), 281-292] who showed that the algebraic Eigen-space can be chosen to consist of a basis each with nonnegative components. This result was applied in a clever way to provide necessary and sufficient conditions concerning solvability of nonnegative matrix equations by S. Friedland and H. Schneider and by H.D. Victory, Jr. [SIAM Journal on Algebraic and Discrete Methods 1 (1980), 185-200; respectively, 6 (1985), 406-412]. The work by U. Rothblum was extended to the setting of eventually compact, nonnegative integral operators on Lp-space, p > 1, by H.D. Victory, Jr. [Journal of Mathematical Analysis and Applications 90 (1982), 484-516]. This thesis consists of two results extending the work by H.D. Victory, Jr., cited in the preceding paragraph. The first portion of this thesis provides necessary and sufficient conditions for nonnegative integral operator equations of the form ëf = Kf + g to possess a nonnegative solution f in Lp when ë > 0 and g is a given and nonnegative element in Lp, p > 1. The second portion studies the lattice ideal structure of the algebraic Eigen-space of an eventually compact positive linear operator belonging to its spectral radius.