Abstract
This work discusses the contributions of the authors to the analysis of nonnegative reducible operators on Banach lattices. In the first portion, we let (Ω, Σ, μ) denote a σ-finite measure space and L p (Ω, Σ,μ) (1 ≤ p < ∞) the usual Banach lattice of real-valued p th summable functions. Suppose, moreover, K is an integral operator with nonnegative kernel, mapping L p (Ω, Σ, μ) into itself, while possessing a compact iterate. We give necessary and sufficient conditions (Theorem 3.6) for the integral operator equation λf = K f + g to possess a nonnegative solution f ∈ L p (Ω, Σ, μ), where 0 ≤ g ∈ L p (Ω, Σ, μ) and λ > 0. In the second half of this work, we study the structure of the algebraic eigenspace corresponding to the spectral radius of a nonnegative reducible linear operator having a completely continuous iterate and defined on a Banach lattice E with order continuous norm. The combinatorial characterization of the Riesz index of the spectral radius and of the dimension of the algebraic eigenspace is made possible by a decomposition of the underlying operator in a form generalizing the Frobenius normal form of a nonnegative reducible matrix.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
C. D. Aliprantis and O. Burkinshaw, Positive Operators, Academic Press, New York & London, 1985.
T. Andô, Positive operators in semi-ordered linear spaces, J. Fac. Sci. Hokkaido Univ., Ser. 1, 13 (1957), 214–228.
G. Birkhoff, Extensions of Jentzsch’s Theorem, Trans. Amer. Math. Soc. 85 (1957), 219–227.
F. F. Bonsall and B. J. Tomiuk, The semi-algebra generated by a compact linear operator, Proc. Edinburgh Math. Soc. 14 (S-II) (1965), 177–196.
C. D. H. Cooper, On the maximum eigenvalue of a reducible nonnegative real matrix, Math. Z. 131 (1973), 213–217.
S. Friedland and H. Schneider, The growth of powers of a nonnegative matrix, SIAM J. Algebraic Discrete Methods 1 (1980), 185–200.
G. Frobenius, Über Matrizen aus nicht-negativen Elementen, Sitz. Berichte Kgl. Preuβ. Akad. Wiss. Berlin, 456–477, 1912.
F. R. Gantmacher, The Theory of Matrices, Vol. II, Gosud. Izdat. Techn.-Theor. Lit., Moscow. (Translated by K. Hirsch, Chelsea, New York, 1959.)
Ruey-Jen Jang, The Ideal Structure of the Algebraic Eigenspace to the Spectral Radius of Eventually Compact, Reducible Positive Linear Operators, Ph.D. Dissertation, Texas Tech University, August, 1990.
R. Jang and H. D. Victory, Jr., On nonnegative solvability of linear integral equations, preprint. (Submitted for publication.)
R. Jang and H. D. Victory, Jr., On the ideal structure of positive, eventually compact linear operators on Banach lattices, preprint. (Submitted for publication.)
P. Jentzsch, Über Integralgleichungen mit positivem Kern, J. Reine Angew. Math. 141 (1912), 235–244.
S. Karlin, Positive operators, J. Math. Mech. 8 (1959), 907–937.
S. Karlin, A First Course in Stochastic Processes, Academic Press, New York, San Francisco, London, 1975.
M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space. Uspekhi Mat. Nauk (N.S.) 3, No. 1 (23), 3–95 (1948) (Russian). Also, Transl. Amer. Math. Soc. # 26, 1950.
H. P. Lotz, Über das Spektrum positiver Operatoren, Math. Z. 108 (1968), 15–32.
H. P. Lotz and H. H. Schaefer, Über einen Satz von F. Niiro und I. Sawashima, Math. Z. 108 (1968), 33–36.
W. A. J. Luxemburg and A. C. Zaanen, Notes on Banach function spaces, XI, Indag. Math. 26 (1964), 507–518.
W. A. J. Luxemburg and A. C. Zaanen, Notes on Banach function spaces, XIV, Indag. Math. 27 (1965), 230–248.
P. Nelson, Jr., The structure of a positive linear integral operator, J. London Math. Soc. (2), 8 (1974), 711–718.
F. Niiro and I. Sawashima, On spectral properties of positive irreducible operators in an arbitrary Banach lattice and problems of H. H. Schaefer, Sci. Papers College Arts Sci. Univ. Tokyo 16 (1966), 145–183.
B. de Pagter, Irreducible compact operators, Math. Z. 192 (1986), 149–153.
O. Perron, Zur Theorie der Matrizen, Math. Ann. 64 (1907), 248–263.
U. G. Rothblum, Algebraic eigenspaces of nonnegative matrices, Linear Algebra Appl. 12 (1975), 281–292.
H. H. Schaefer, Some spectral properties of positive linear operators, Pacific J. Math. 10 (1960), 1009–1019.
H. H. Schaefer, Banach Lattices and Positive Operators, Springer-Verlag, New York, Heidelberg, Berlin, 1974.
R. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, N. J., 1962.
H. D. Victory, Jr., On linear integral operators with nonnegative kernels, J. Math. Anal. Appl. 89 (1982), 420–441.
H. D. Victory, Jr., The structure of the algebraic eigenspace to the spectral radius of eventually compact, nonnegative integral operators, J. Math. Anal. Appl. 90 (1982), 484–516.
H. D. Victory, Jr., On nonnegative solutions of matrix equations, SIAM J. Algebraic Discrete Methods 6 (1985), 406–412.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Jang, RJ., Victory, H.D. (1991). Frobenius Decomposition of Positive Compact Operators. In: Positive Operators, Riesz Spaces, and Economics. Studies in Economic Theory, vol 2. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58199-1_11
Download citation
DOI: https://doi.org/10.1007/978-3-642-58199-1_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-63502-1
Online ISBN: 978-3-642-58199-1
eBook Packages: Springer Book Archive