77 research outputs found
Completability and optimal factorization norms in tensor products of Banach function spaces
[EN] Given s-finite measure spaces ( 1, 1, mu 1) and ( 2, 2, mu 2), we consider Banach spaces X1(mu 1) and X2(mu 2), consisting of L0(mu 1) and L0(mu 2) measurable functions respectively, and study when the completion of the simple tensors in the projective tensor product X1(mu 1). p X2(mu 2) is continuously included in the metric space of measurable functions L0(mu 1. mu 2). In particular, we prove that the elements of the completion of the projective tensor product of L p-spaces are measurable functions with respect to the product measure. Assuming certain conditions, we finally showthat given a bounded linear operator T : X1(mu 1). p X2(mu 2). E (where E is a Banach space), a norm can be found for T to be bounded, which is ` minimal' with respect to a given property (2-rectangularity). The same technique may work for the case of n-spaces.J. M. Calabuig and M. Fernandez-Unzueta were supported by Ministerio de Economia, Industria y Competitividad (Spain) under project MTM2014-53009-P. M. Fernandez-Unzueta was also suported by CONACyT 284110. F. Galaz-Fontes was supported by Ministerio de Ciencia e Innovacion (Spain) and FEDER under project MTM2009-14483-C02-01. E. A. Sanchez Perez was supported by Ministerio de Economia, Industria y Competitividad (Spain) and FEDER under project MTM2016-77054-C2-1-P.Calabuig, JM.; FernĂĄndez-Unzueta, M.; Galaz-Fontes, F.; SĂĄnchez PĂ©rez, EA. (2019). Completability and optimal factorization norms in tensor products of Banach function spaces. Revista de la Real Academia de Ciencias Exactas FĂsicas y Naturales Serie A MatemĂĄticas. 113(4):3513-3530. https://doi.org/10.1007/s13398-019-00711-7S351335301134Abramovich, Y.A., Aliprantis, C.D.: An invitation to operator theory. Graduate Studies in Mathematics, Vol 50, AMS (2002)Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Boston (1988)Bu, Q., Buskes, G., Kusraev, A.G.: Bilinear maps on products of vector lattices: a survey. In: Boulabiar, K., Buskes, G., Triki, A. (eds.) Positivity-Trends in Mathematics. BirkhĂ€ser Verlag AG, Basel, pp. 97â26 (2007)Buskes, G., Van Rooij, A.: Bounded variation and tensor products of Banach lattices. Positivity 7, 47â59 (2003)Calabuig, J.M., FernĂĄndez-Unzueta, M., Galaz-Fontes, F., SĂĄnchez-PĂ©rez, E.A.: Extending and factorizing bounded bilinear maps defined on order continuous Banach function spaces. RACSAM 108(2), 353â367 (2014)Calabuig, J.M., FernĂĄndez-Unzueta, M., Galaz-Fontes, F., SĂĄnchez-PĂ©rez, E.A.: Equivalent norms in a Banach function space and the subsequence property. J. Korean Math. Soc. https://doi.org/10.4134/JKMS.j180682Curbera, G.P., Ricker, W.J.: Optimal domains for kernel operators via interpolation. Math. Nachr. 244, 47â63 (2002)Curbera, G.P., Ricker, W.J.: Vector measures, integration and applications. In: Positivity. BirkhĂ€user Basel, pp. 127â160 (2007)Gil de Lamadrid, J.: Uniform cross norms and tensor products. J. Duke Math. 32, 797â803 (1965)Dunford, N., Schwartz, J.: Linear Operators, Part I: General Theory. Interscience Publishers Inc., New York (1958)Fremlin, D.H.: Tensor products of Archimedean vector lattices. Am. J. Math. 94(3), 777â798 (1972)Fremlin, D.H.: Tensor products of Banach lattices. Math. Ann. 211(2), 87â106 (1974)Yew, K.L.: Completely -summing maps on the operator Hilbert space OH. J. Funct. Anal. 255, 1362â1402 (2008)Kwapien, S., Pelczynski, A.: The main triangle projection in matrix spaces and its applications. Stud. Math. 34(1), 43â68 (1970)Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces II. Springer, Berlin (1979)Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces I. North-Holland Publishing Company, Amsterdam (1971)Milman, M.: Some new function spaces and their tensor products. Depto. de MatemĂĄtica, Facultad de Ciencias, U. de los Andes, MĂ©rida, Venezuela (1978)Okada, S., Ricker, W.J., SĂĄnchez PĂ©rez, E.A.: Optimal domain and integral extension of operators acting in function spaces. Oper. Theory Adv. Appl., vol. 180. BirkhĂ€user, Basel (2008)Schep, A.R.: Factorization of positive multilinear maps. Illinois J. Math. 579â591 (1984)Zaanen, A.C.: Integration. North-Holland Publishing Company, Amsterdam-New York (1967)Zaanen, A.C.: Riesz Spaces II. North-Holland Publishing Company, Amsterdam (1983
Constructive pointfree topology eliminates non-constructive representation theorems from Riesz space theory
In Riesz space theory it is good practice to avoid representation theorems
which depend on the axiom of choice. Here we present a general methodology to
do this using pointfree topology. To illustrate the technique we show that
almost f-algebras are commutative. The proof is obtained relatively
straightforward from the proof by Buskes and van Rooij by using the pointfree
Stone-Yosida representation theorem by Coquand and Spitters
Neutron Scattering and Its Application to Strongly Correlated Systems
Neutron scattering is a powerful probe of strongly correlated systems. It can
directly detect common phenomena such as magnetic order, and can be used to
determine the coupling between magnetic moments through measurements of the
spin-wave dispersions. In the absence of magnetic order, one can detect diffuse
scattering and dynamic correlations. Neutrons are also sensitive to the
arrangement of atoms in a solid (crystal structure) and lattice dynamics
(phonons). In this chapter, we provide an introduction to neutrons and neutron
sources. The neutron scattering cross section is described and formulas are
given for nuclear diffraction, phonon scattering, magnetic diffraction, and
magnon scattering. As an experimental example, we describe measurements of
antiferromagnetic order, spin dynamics, and their evolution in the
La(2-x)Ba(x)CuO(4) family of high-temperature superconductors.Comment: 31 pages, chapter for "Strongly Correlated Systems: Experimental
Techniques", edited by A. Avella and F. Mancin
On the Banach lattice structure of L-w(1) of a vector measure on a delta-ring
We study some Banach lattice properties of the space L-w(1)(v) of weakly integrable functions with respect to a vector measure v defined on a delta-ring. Namely, we analyze order continuity, order density and Fatou type properties. We will see that the behavior of L-w(1)(v) differs from the case in which is defined on a sigma-algebra whenever does not satisfy certain local sigma-finiteness property.J. M. Calabuig and M. A. Juan were supported by the Ministerio de Economia y Competitividad (project MTM2008-04594). O. Delgado was supported by the Ministerio de Economia y Competitividad (project MTM2009-12740-C03-02). E. A. Sanchez Perez was supported by the Ministerio de Economia y Competitividad (project MTM2009-14483-C02-02).Calabuig Rodriguez, JM.; Delgado Garrido, O.; Juan Blanco, MA.; SĂĄnchez PĂ©rez, EA. (2014). On the Banach lattice structure of L-w(1) of a vector measure on a delta-ring. Collectanea Mathematica. 65(1):67-85. doi:10.1007/s13348-013-0081-8S6785651Brooks, J.K., Dinculeanu, N.: Strong additivity, absolute continuity and compactness in spaces of measures. J. Math. Anal. Appl. 45, 156â175 (1974)Calabuig, J.M., Delgado, O., SĂĄnchez PĂ©rez, E.A.: Factorizing operators on Banach function spaces through spaces of multiplication operators. J. Math. Anal. Appl. 364, 88â103 (2010)Calabuig, J.M., Juan, M.A., SĂĄnchez PĂ©rez, E.A.: Spaces of -integrable functions with respect to a vector measure defined on a -ring. Oper. Matrices 6, 241â262 (2012)Curbera, G.P.: El espacio de funciones integrables respecto de una medida vectorial. Ph. D. thesis, University of Sevilla, Sevilla (1992)Curbera, G.P.: Operators into of a vector measure and applications to Banach lattices. Math. Ann. 293, 317â330 (1992)Curbera, G.P., Ricker, W.J.: Banach lattices with the Fatou property and optimal domains of kernel operators. Indag. Math. (N.S.) 17, 187â204 (2006)G. P. Curbera and W. J. Ricker, Vector measures, integration and applications. In: Positivity (in Trends Math.), BirkhĂ€user, Basel, pp. 127â160 (2007)Curbera, G.P., Ricker, W.J.: The Fatou property in -convex Banach lattices. J. Math. Anal. Appl. 328, 287â294 (2007)Delgado, O.: -spaces of vector measures defined on -rings. Arch. Math. 84, 432â443 (2005)Delgado, O.: Optimal domains for kernel operators on . Studia Math. 174, 131â145 (2006)Delgado, O., Soria, J.: Optimal domain for the Hardy operator. J. Funct. Anal. 244, 119â133 (2007)Delgado, O., Juan, M.A.: Representation of Banach lattices as spaces of a vector measure defined on a -ring. Bull. Belg. Math. Soc. Simon Stevin 19(2), 239â256 (2012)Diestel, J., Uhl, J.J.: Vector measures (Am. Math. Soc. surveys 15). American Mathematical Society, Providence (1997)Dinculeanu, N.: Vector measures, Hochschulbcher fr Mathematik, vol. 64. VEB Deutscher Verlag der Wissenschaften, Berlin (1966)FernĂĄndez, A., Mayoral, F., Naranjo, F., SĂĄez, C., SĂĄnchez PĂ©rez, E.A.: Spaces of -integrable functions with respect to a vector measure. Positivity 10, 1â16 (2006)Fremlin, D.H.: Measure theory, broad foundations, vol. 2. Torres Fremlin, Colchester (2001)JimĂ©nez FernĂĄndez, E., Juan, M.A., SĂĄnchez PĂ©rez, E.A.: A KomlĂłs theorem for abstract Banach lattices of measurable functions. J. Math. Anal. Appl. 383, 130â136 (2011)Lewis, D.R.: On integrability and summability in vector spaces. Ill. J. Math. 16, 294â307 (1972)Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces II. Springer, Berlin (1979)Luxemburg, W.A.J., Zaanen, A.C.: Riesz spaces I. North-Holland, Amsterdam (1971)Masani, P.R., Niemi, H.: The integration theory of Banach space valued measures and the Tonelli-Fubini theorems. I. Scalar-valued measures on -rings. Adv. Math. 73, 204â241 (1989)Masani, P.R., Niemi, H.: The integration theory of Banach space valued measures and the Tonelli-Fubini theorems. II. Pettis integration. Adv. Math. 75, 121â167 (1989)Thomas, E.G.F.: Vector integration (unpublished) (2013)Turpin, Ph.: IntĂ©gration par rapport Ă une mesure Ă valeurs dans un espace vectoriel topologique non supposĂ© localement convexe, IntĂšgration vectorielle et multivoque, (Colloq., University Caen, Caen, 1975), experiment no. 8, DĂšp. Math., UER Sci., University Caen, Caen (1975)Okada, S., Ricker, W.J., SĂĄnchez PĂ©rez, E.A.: Optimal domain and integral extension of operators acting in function spaces (Oper. Theory Adv. Appl.), vol. 180. BirkhĂ€user, Basel (2008)Zaanen, A.C.: Riesz spaces II. North-Holland, Amsterdam (1983
Kothe dual of Banach lattices generated by vector measures
We study the Kothe dual spaces of Banach function lattices generated by abstract methods having roots in the theory of interpolation spaces. We apply these results to Banach spaces of integrable functions with respect to Banach space valued countably additive vector measures. As an application we derive a description of the Banach dual of a large class of these spaces, including Orlicz spaces of integrable functions with respect to vector measuresThe first author was supported by the Foundation for Polish Science (FNP). The second author was supported by the Ministerio de Economia y Competitividad (Spain) under Grant #MTM2012-36740-C02-02.Mastylo, M.; SĂĄnchez PĂ©rez, EA. (2014). Kothe dual of Banach lattices generated by vector measures. Monatshefte fur Mathematik. 173(4):541-557. https://doi.org/10.1007/s00605-013-0560-8S5415571734Aronszajn, N., Gagliardo, E.: Interpolation spaces and interpolation methods. Ann. Mat. Pura. Appl. 68, 51â118 (1965)Bartle, R.G., Dunford, N., Schwartz, J.: Weak compactness and vector measures. Canad. J. Math. 7, 289â305 (1955)Brudnyi, Yu.A., Krugljak, N.Ya.: Interpolation functors and interpolation spaces I . North-Holland, Amsterdam (1991)Curbera, G.P.: Operators into L 1 of a vector measure and applications to Banach lattices. Math. Ann. 293, 317â330 (1992)Curbera, G.P., Ricker, W.J.: The Fatou property in p -convex Banach lattices. J. Math. Anal. Appl. 328, 287â294 (2007)Delgado, O.: Banach function subspaces of L 1 of a vector measure and related Orlicz spaces. Indag. Math. 15(4), 485â495 (2004)Diestel, J., Jr., Uhl, J.J.: Vector measures, Amer. Math. Soc. Surveys 15, Providence, R.I. (1977)FernĂĄndez, A., Mayoral, F., Naranjo, F., SĂĄnchez-PĂ©rez, E.A.: Spaces of p -integrable functions with respect to a vector measure. Positivity 10, 1â16 (2006)Ferrando, I., RodrĂguez, J.: The weak topology on L p of a vector measure. Topol. Appl. 155, 1439â1444 (2008)Ferrando, I., SĂĄnchez PĂ©rez, E.A.: Tensor product representation of the (pre)dual of the L p -space of a vector measure. J. Aust. Math. Soc. 87, 211â225 (2009)Galaz-Fontes, F.: The dual space of L p of a vector measure. Positivity 14(4), 715â729 (2010)KamiĆska, A.: Indices, convexity and concavity in Musielak-Orlicz spaces, dedicated to Julian Musielak. Funct. Approx. Comment. Math. 26, 67â84 (1998)Kantorovich, L.V., Akilov, G.P.: Functional analysis, 2nd edn. Pergamon Press, New York (1982)Krein, S.G., Petunin, Yu.I., Semenov, E.M.: Interpolation of linear operators. In: Translations of mathematical monographs, 54. American Mathematical Society, Providence, R.I., (1982)Lewis, D.R.: Integration with respect to vector measures. Pacific. J. Math. 33, 157â165 (1970)Lewis, D.R.: On integrability and summability in vector spaces. Ill. J. Math. 16, 583â599 (1973)Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces II. Springer, Berlin (1979)Lozanovskii, G.Ya.: On some Banach lattices, (Russian). Sibirsk. Mat. Z. 10, 419â430 (1969)Musielak, J.: Orlicz spaces and modular spaces. In: Lecture Notes in Math. 1034, Springer-Verlag, Berlin (1983)Okada, S.: The dual space of L 1 ( ÎŒ ) of a vector measure ÎŒ . J. Math. Anal. Appl. 177, 583â599 (1993)Okada, S., Ricker, W.J., SĂĄnchez PĂ©rez, E.A.: Optimal domain and integral extension of operators acting in function spaces, operator theory. Adv. Appl., vol. 180, BirkhĂ€user, Basel (2008)Rao, M.M., Zen, Z.D.: Applications of Orlicz spaces. Marcel Dekker, Inc., New York (2002)Rivera, M.J.: Orlicz spaces of integrable functions with respect to vector-valued measures. Rocky Mt. J. Math. 38(2), 619â637 (2008)SĂĄnchez PĂ©rez, E.A.: Compactness arguments for spaces of p -integrable functions with respect to a vector measure and factorization of operators through Lebesgue-Bochner spaces. Ill. J. Math. 45(3), 907â923 (2001)SĂĄnchez PĂ©rez, E.A.: Vector measure duality and tensor product representation of L p spaces of vector measures. Proc. Amer. Math. Soc. 132, 3319â3326 (2004)Zaanen, A.C.: Integration. North Holland, Amsterdam (1967
IL-21 and IL-6 Are Critical for Different Aspects of B Cell Immunity and Redundantly Induce Optimal Follicular Helper CD4 T Cell (Tfh) Differentiation
Cytokines are important modulators of lymphocytes, and both interleukin-21 (IL-21) and IL-6 have proposed roles in T follicular helper (Tfh) differentiation, and directly act on B cells. Here we investigated the absence of IL-6 alone, IL-21 alone, or the combined lack of IL-6 and IL-21 on Tfh differentiation and the development of B cell immunity in vivo. C57BL/6 or IL-21â/â mice were treated with a neutralizing monoclonal antibody against IL-6 throughout the course of an acute viral infection (lymphocytic choriomeningitis virus, LCMV). The combined absence of IL-6 and IL-21 resulted in reduced Tfh differentiation and reduced Bcl6 protein expression. In addition, we observed that these cytokines had a large impact on antigen-specific B cell responses. IL-6 and IL-21 collaborate in the acute T-dependent antiviral antibody response (90% loss of circulating antiviral IgG in the absence of both cytokines). In contrast, we observed reduced germinal center formation only in the absence of IL-21. Absence of IL-6 had no impact on germinal centers, and combined absence of both IL-21 and IL-6 revealed no synergistic effect on germinal center B cell development. Studying CD4 T cells in vitro, we found that high IL-21 production was not associated with high Bcl6 or CXCR5 expression. TCR stimulation of purified naĂŻve CD4 T cells in the presence of IL-6 also did not result in Tfh differentiation, as determined by Bcl6 or CXCR5 protein expression. Cumulatively, our data indicates that optimal Tfh formation requires IL-21 and IL-6, and that cytokines alone are insufficient to drive Tfh differentiation
Maurey-Rosenthal domination for abstract Banach lattices
We extend the Maurey-Rosenthal theorem on integral domination and factorization
of p-concave operators from a p-convex Banach function space through Lp-spaces for
the case of operators on abstract p-convex Banach lattices satisfying some essential
lattice requirements - mainly order density of its order continuous part - that are
shown to be necessary. We prove that these geometric properties can be
characterized by means of an integral inequality giving a domination of the pointwise
evaluation of the operator for a suitable weight also in the case of abstract Banach
lattices. We obtain in this way what in a sense can be considered the most general
factorization theorem of operators through Lp-spaces. In order to do this, we prove a
new representation theorem for abstract p-convex Banach lattices with the Fatou
property as spaces of p-integrable functions with respect to a vector measure.The authors are supported by grants MTM2011-23164 and MTM2012-36740-C02-02 of the Ministerio de Economia y Competitividad (Spain).Juan Blanco, MA.; SĂĄnchez PĂ©rez, EA. (2013). Maurey-Rosenthal domination for abstract Banach lattices. Journal of Inequalities and Applications. (213). https://doi.org/10.1186/1029-242X-2013-213S213Defant A: Variants of the Maurey-Rosenthal theorem for quasi Köthe function spaces. Positivity 2001, 5: 153â175. 10.1023/A:1011466509838Defant A, SĂĄnchez PĂ©rez EA: Maurey-Rosenthal factorization of positive operators and convexity. J. Math. Anal. Appl. 2004, 297: 771â790. 10.1016/j.jmaa.2004.04.047Defant A, SĂĄnchez PĂ©rez EA: Domination of operators on function spaces. Math. Proc. Camb. Philos. Soc. 2009, 146: 57â66. 10.1017/S0305004108001734FernĂĄndez A, Mayoral F, Naranjo F, SĂĄez C, SĂĄnchez-PĂ©rez EA: Vector measure Maurey-Rosenthal type factorizations and l -sums of L 1 -spaces. J. Funct. Anal. 2005, 220: 460â485. 10.1016/j.jfa.2004.06.010Palazuelos C, SĂĄnchez PĂ©rez EA, Tradacete P: Maurey-Rosenthal factorization for p -summing operators and Dodds-Fremlin domination. J. Oper. Theory 2012, 68(1):205â222.Luxemburg WAJ, Zaanen AC: Riesz Spaces I. North-Holland, Amsterdam; 1971.Zaanen AC: Riesz Spaces II. North-Holland, Amsterdam; 1983.Lindenstrauss J, Tzafriri L: Classical Banach Spaces II. Springer, Berlin; 1979.Aliprantis CD, Burkinshaw O: Positive Operators. Academic Press, New York; 1985.Curbera GP, Ricker WJ: Vector measures, integration and applications. Trends Math. In Positivity. BirkhĂ€user, Basel; 2007:127â160.Okada S, Ricker WJ, SĂĄnchez PĂ©rez EA: Optimal domains and integral extensions of operators acting in function spaces. 180. In Operator Theory Advances and Applications. BirkhĂ€user, Basel; 2008.Delgado O: L 1 -spaces of vector measures defined on ÎŽ -rings. Arch. Math. 2005, 84: 432â443. 10.1007/s00013-005-1128-1Calabuig, JM, Delgado, O, Juan, MA, SĂĄnchez PĂ©rez, EA: On the Banach lattice structure of L w 1 of a vector measure on a ÎŽ-ring. Collect. Math. doi:10.1007/s13348â013â0081â8Calabuig JM, Delgado O, SĂĄnchez PĂ©rez EA: Factorizing operators on Banach function spaces through spaces of multiplication operators. J. Math. Anal. Appl. 2010, 364: 88â103. 10.1016/j.jmaa.2009.10.034Delgado O:Optimal domains for kernel operators on [ 0 , â ) Ă [ 0 , â ) .Stud. Math. 2006, 174: 131â145. 10.4064/sm174-2-2Delgado O, Soria J: Optimal domain for the Hardy operator. J. Funct. Anal. 2007, 244: 119â133. 10.1016/j.jfa.2006.12.011JimĂ©nez FernĂĄndez E, Juan MA, SĂĄnchez PĂ©rez EA: A KomlĂłs theorem for abstract Banach lattices of measurable functions. J. Math. Anal. Appl. 2011, 383: 130â136. 10.1016/j.jmaa.2011.05.010Curbera, GP: El espacio de funciones integrables respecto de una medida vectorial. PhD thesis, Univ. of Sevilla (1992)SĂĄnchez PĂ©rez EA: Compactness arguments for spaces of p -integrable functions with respect to a vector measure and factorization of operators through Lebesgue-Bochner spaces. Ill. J. Math. 2001, 45(3):907â923.FernĂĄndez A, Mayoral F, Naranjo F, SĂĄez C, SĂĄnchez-PĂ©rez EA: Spaces of p -integrable functions with respect to a vector measure. Positivity 2006, 10: 1â16. 10.1007/s11117-005-0016-zCalabuig JM, Juan MA, SĂĄnchez PĂ©rez EA: Spaces of p -integrable functions with respect to a vector measure defined on a ÎŽ -ring. Oper. Matrices 2012, 6: 241â262.Lewis DR: On integrability and summability in vector spaces. Ill. J. Math. 1972, 16: 294â307.Masani PR, Niemi H: The integration theory of Banach space valued measures and the Tonelli-Fubini theorems. I. Scalar-valued measures on ÎŽ -rings. Adv. Math. 1989, 73: 204â241. 10.1016/0001-8708(89)90069-8Masani PR, Niemi H: The integration theory of Banach space valued measures and the Tonelli-Fubini theorems. II. Pettis integration. Adv. Math. 1989, 75: 121â167. 10.1016/0001-8708(89)90035-2Brooks JK, Dinculeanu N: Strong additivity, absolute continuity and compactness in spaces of measures. J. Math. Anal. Appl. 1974, 45: 156â175. 10.1016/0022-247X(74)90130-9Curbera GP:Operators into L 1 of a vector measure and applications to Banach lattices.Math. Ann. 1992, 293: 317â330. 10.1007/BF01444717Delgado O, Juan MA: Representation of Banach lattices as L w 1 spaces of a vector measure defined on a ÎŽ -ring. Bull. Belg. Math. Soc. Simon Stevin 2012, 19: 239â256.Curbera GP, Ricker WJ: Banach lattices with the Fatou property and optimal domains of kernel operators. Indag. Math. 2006, 17: 187â204. 10.1016/S0019-3577(06)80015-7Curbera GP, Ricker WJ: The Fatou property in p -convex Banach lattices. J. Math. Anal. Appl. 2007, 328: 287â294. 10.1016/j.jmaa.2006.04.086Aliprantis CD, Border KC: Infinite Dimensional Analysis. Springer, Berlin; 1999.Delgado, O: Optimal extension for positive order continuous operators on Banach function spaces. Glasg. Math. J. (to appear
The one dimensional Kondo lattice model at partial band filling
The Kondo lattice model introduced in 1977 describes a lattice of localized
magnetic moments interacting with a sea of conduction electrons. It is one of
the most important canonical models in the study of a class of rare earth
compounds, called heavy fermion systems, and as such has been studied
intensively by a wide variety of techniques for more than a quarter of a
century. This review focuses on the one dimensional case at partial band
filling, in which the number of conduction electrons is less than the number of
localized moments. The theoretical understanding, based on the bosonized
solution, of the conventional Kondo lattice model is presented in great detail.
This review divides naturally into two parts, the first relating to the
description of the formalism, and the second to its application. After an
all-inclusive description of the bosonization technique, the bosonized form of
the Kondo lattice hamiltonian is constructed in detail. Next the
double-exchange ordering, Kondo singlet formation, the RKKY interaction and
spin polaron formation are described comprehensively. An in-depth analysis of
the phase diagram follows, with special emphasis on the destruction of the
ferromagnetic phase by spin-flip disorder scattering, and of recent numerical
results. The results are shown to hold for both antiferromagnetic and
ferromagnetic Kondo lattice. The general exposition is pedagogic in tone.Comment: Review, 258 pages, 19 figure
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