Riesz空间中特殊算子及其逆算子性质研究
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摘要
在有关Riesz空间及其上的算子理论的研究中,主要集中在空间性质和算子性质两个重要方面。空间性质则主要讨论空间的拓扑结构,序结构,以及两者之间的相容性等问题；算子性质主要讨论算子的控制性、分解性、格序性、和算子与算子之间的关系,以及算子空间的空间性质等问题。目前有关算子性质研究有相当重要的结果都主要集中在特殊算子类,如保不交算子,带算子,序理想算子以及紧算子等。特殊算子的性质研究是Riesz空间及其上的算子理论的重要研究内容。本文就Riesz空间中几类特殊算子及其逆算子的性质做了较为深入的研究,主要内容分为以下三个部分：
第一部分研究了Riesz空间中特殊算子的性质。主要涉及到三个方面：一是,较为系统的讨论了保不交算子和序理想算子的性质；二是,讨论了带算子的性质,并研究了带算子与保不交等算子之间关系；三是,利用带算子的性质讨论了与保不交算子存在着紧密联系的一些条件之间的等价性问题。
第二部分综述了保不交等特殊算子的逆算子的性质,讨论了在什么条件下其逆算子的遗传性质成立。类似的,研究了带算子的逆算子性质。得到了在一般情况下可逆带算子其逆算子不一定是带算子的结论,并给出了一些附加条件使得可逆的带算子其逆算子仍是带算子。
第三部分研究了带算子等特殊算子的正则性。首先给出了一个反例说明了并不是所有的带算子都是正则的,其次讨论了在什么情况下带算子是正则的,最后给出了带算子是正则算子的充要条件。
The operator property is a famous problem in the history of operator theory. This paper studies the nature of some kinds of special operators and their inverse of operators in the Riese space, mainly it has three parts:
In the first part, we investigate the nature of the special operators in the Riese space. First, based on the outcome of the preceding research, it discusses the nature of the disjointness preserving operator. Then, the nature of the band operator and studies the relationship between band operator and disjointness preserving operator. at last, discussing some equal problems for some conditions which have very close relations with the disjointness preserving operator.
The second part is devoted to the nature of the inverse of operator for the disjointness preserving operator and so on. It discusses that the inverse of operator is still of this kind of nature under what kind of conditions. Similarly, we have obtained the result that inverse of operator must not be band operator under general condition by studying the nature of inverse of operator of band operator. At the same time providing some weak conditions which make inverse of operator is still band operator.
Finally, we researched the regularity of special operators such as band operator and so on. first of all, it gives an example that explains not all of band operators are regular, secondly, it discusses that under what certain condition band operator is regular, at last it gives us that band operator is regular operator indeed.
第一部分研究了Riesz空间中特殊算子的性质。主要涉及到三个方面：一是,较为系统的讨论了保不交算子和序理想算子的性质；二是,讨论了带算子的性质,并研究了带算子与保不交等算子之间关系；三是,利用带算子的性质讨论了与保不交算子存在着紧密联系的一些条件之间的等价性问题。
第二部分综述了保不交等特殊算子的逆算子的性质,讨论了在什么条件下其逆算子的遗传性质成立。类似的,研究了带算子的逆算子性质。得到了在一般情况下可逆带算子其逆算子不一定是带算子的结论,并给出了一些附加条件使得可逆的带算子其逆算子仍是带算子。
第三部分研究了带算子等特殊算子的正则性。首先给出了一个反例说明了并不是所有的带算子都是正则的,其次讨论了在什么情况下带算子是正则的,最后给出了带算子是正则算子的充要条件。
The operator property is a famous problem in the history of operator theory. This paper studies the nature of some kinds of special operators and their inverse of operators in the Riese space, mainly it has three parts:
In the first part, we investigate the nature of the special operators in the Riese space. First, based on the outcome of the preceding research, it discusses the nature of the disjointness preserving operator. Then, the nature of the band operator and studies the relationship between band operator and disjointness preserving operator. at last, discussing some equal problems for some conditions which have very close relations with the disjointness preserving operator.
The second part is devoted to the nature of the inverse of operator for the disjointness preserving operator and so on. It discusses that the inverse of operator is still of this kind of nature under what kind of conditions. Similarly, we have obtained the result that inverse of operator must not be band operator under general condition by studying the nature of inverse of operator of band operator. At the same time providing some weak conditions which make inverse of operator is still band operator.
Finally, we researched the regularity of special operators such as band operator and so on. first of all, it gives an example that explains not all of band operators are regular, secondly, it discusses that under what certain condition band operator is regular, at last it gives us that band operator is regular operator indeed.
引文
[1]Y. A. Abramovich, A. I. Veksler and A. V. Koldunov. Operators preserving disjointness[J]. Dokl. Akad. Nauk SSSR 248 (1979),1033-1036; English transl., Soviet Math. Dokl.20(1979),1089-1093. MR 81e:47034.
[2]Zaanen, A.C. Riesz spaces II[M]. North-Holland, Amsterdam,1983.
[3]Y. A. Abramovich, A. K. Kitover. A solution to a problem on invertible disjointness preserving operators[J]. Proc. Amer. Math. Soc.126 (1998),1501-1505.
[4]Huijsmans, C.B, De Pagter, B. Intertible disjointness preserving operators[J]. Proc. of the Edinburgh Mat. Soc.37 (1993),125-132.
[5]Y. A. Abramovich, A. K. Kitover. Inverses of Disjointness Preserving Operators[J]. Memoirs of the Amer. Math. Soc.,143(2000), No.679.
[6]Turan, B. On Ideal Operators[J]. Positivity, (2003),7(1-2),141-148.
[7]Meyer-Nieberg, P. Banach lattices[M]. Springer-Verlag, Berlin, Heideberg, New York,1991.
[8]Y. A. Abramovich and A. K. Kitover. A characterization of operators preserving disjointness in terms of their inverse[J]. Positivity,4(2000),205-212.
[9]C. B. Huijsmans, W. A. J. Luxemburg. Positive Operators and Semigroups on Banach Lattices[J]. Kluwer, Dordrecht,1992. MR 93e:47003.
[10]Y. A. Abramovich. Multiplicative representation of disjointness preserving operators [J]. Netherl. Acad. Wetensch. Proc. Ser. A 86 (1983),265-279. M85f:47040.
[11]Ayse Uyar. Solutions of Two Problems in the Theory of Disjointness Preserving Operators[J]. Positivity 11 (2007),119-121.
[12]Y. A. Abramovich, A. K. Kitover. d-independence and d-bases in vector lattices[J]. Positivity 7 (2003),95-97.
[13]C.B.Huijsmans, B.Depagter, Disjointness preserving and diffuse operator[J]. Compsitio Mathematiea,1991, vol79:351-374.
[14]艾富菊.经典Banach格上保不交算子的性质[D].西南交通大学,2008
[15]C.D.Aliprantis,O.Burkin shaw,Positive Operators[M],Academic Press,INC.,New-York London,1985.
[16]Fethi Benamor,Karim Boulabiar. Maximal ideals of disjointness preserving operators,[J].J Mathematical analysis and applications,2006, Vol322.
[13]C.B.Huijsmans,A.W.Wickstead. The inverse of band preserving and disjointness preserving operators,Indag[J].Mathem.N.S.1992,,vols
[17]Yuri, A. Abramovich, Arkady K.kitover, A characterization of operators preserving disjointness in terms of their inverse [J],Posotovoty,2000,vol 4
[18]陈滋利.Banach格上正则算子格[J].数学学报,2000,(02)
[19]Xiao-dong Zhang,Decomposition theorems for disjointness preserving operators[J],J.Functional Analysis,1993,vol 116.
[20]C.B.Huijsmans, some remarks on disjointness preserving operators [J], Acta Appl.Mathe.,1992.vol 7
[21]W.A.J.Luxemberg,A.C.Zaanen,Riesz Space I [M],North-Holland, Amsterdam New York-Oxford 1971
[23]冯颖,陈滋利,冯天祥,一类经典Banach格上保不交算子的矩阵刻画[J].西南交通大学学报,2003,38卷(4)
[24]A. V. Koldunov, Hammerstein operators preserving disjointness [J], Proceeding of AMS,1995,vol 123
[25]B.De Pagter,A.R.Schep,Band decompositions for disjointness pesering operators [J]Positivaty,2000,vol4
[26]陈滋利.《A Characterization of Non-regular operaters》数学进展,2000,(10)
[27]Adriaan C Zaanen. Introduction to Operator Theory in Riesz Spaces.Berlin-Heidelberg-New York:Springer Verlag,1996.
[28]Koldunov,A. Hammerstein operators preserving disjointness.Proc. Amer. Math. Soc. 1995,123.
[29]Abramovich,Y.A.,Veksler,A.I.,Koldunov,A.V. On operators preserving disjointness.Soviet Math. Dokl.1979.
[30]B.de Pagter. A note on disjointness preserving operator [J]. Proc.Amer.Math.soc. 90(1984).
[31]A.K.Kitover. On disjoint operators in Banach Lattices [J]. Soviet.Math.Dokl250, N4,800-803(1980)
[32]Y.A.Abramovich and A.K.Kitover. Bijective disjointness preserving operator[J]. Functional Analysis and Economic Theorey, Springer,Berlin New York,1998, pp.1-8.
[34]P. T. N. McPolin and A. W. Wickstead, The order boundedness of band preserving operators on uniformly complete vector lattices, Math. Proc. Cambridge Phil. Soc.97 (1985),481-487.
[35]Y. A. Abramovich, E. L. Arenson, A. K. Kitover, Banach C (K)-modules and operators preserving disjointness[J]. Pitman Research Notes in Mathematical Series #277, Longman Scientific & Technical,1992.
[36]W.A.J.Luxemburg, Some Aspects of the Theory of Riesz space[J].Univ. Arkansas Lecture Notes in Math.,4,1979.
[37]Y.A.Abramovich and A.K.Kitover. Inverse and regularity of disjointness preserving operators[J]. Institute of Mathematics,2005.
[38]B.Z.Vulikh. On linear multiplicative operators[J]. Dokl.Akad. Nauk USSR 41(1943), 148-151.
[39]B.Z.Vulikh.Multiplicative in linear semi-order spaces and its application to the theory of operations[J]. Mat.Sbomik 22(1948),267-317.
[40]B.Z.Vulikh. Introduction to the theory of partially ordered spsces[M]. Wolters-Noordhoff Sci.Publication, Groningen,1967.
[41]K.Jarosz. Automatic continuity of separating linear isomorphisms[J]. Canad.Math.Bull.33(1990) 139-144.
[42]Y.A.Abramovich, A.I.Veksler, A.V.Koldunov. Operators preserving disjoitness their continuity and multiplicative representation[J]. Linear operators and their appl. SbornikNauchn. Trudov, Leningrad,1981,13-34.
[43]J.Araujo and J.J.Font. Linear isometries between subspaces of continuous functions[J], Trans.Amer.Math.Soc.349(1997),413-428.
[44]J.J.Font.Automatic continuity of certain isomorphisms between regular Banach function algebras [M]. Glasgo Math.J.39,(1997),333-343.
[45]J.J.Font. Disjointness preserving mappings berween Fourier algebras[J].Proceed. Edinburgh.Math.Soc., submitted.
[46]J.J.Font and S.Hernandez. On separating maps between locally compact spaces[J].Archiv der Math.63(1994),158-165.
[47]J.J.Font and S.Hernandez. Automatic continuity and representation of certain linear isomorphisms between group algebras[J]. Indag. Math.6(1995),397-409.
[48]Z.Ercan, A.W.Wickstead. Toward a theory of non-linear orthomorphisms[J]. Functional Analysis and Economic Theory, Springer, Berlin New York,1998, pp65-87.
[49]A.V.koldunov. Order continuity and representation of operators with the Hammerstein property in vector lattices[J]. Bull.Pol.Acad.Mat.41(1993),43-50.
[50]Y.A.Abramovich and A.K.Kitover. Inverses and regularity of band preserving operators[J]. Indag. Mathem., N.S.,13(2),143-167.
[2]Zaanen, A.C. Riesz spaces II[M]. North-Holland, Amsterdam,1983.
[3]Y. A. Abramovich, A. K. Kitover. A solution to a problem on invertible disjointness preserving operators[J]. Proc. Amer. Math. Soc.126 (1998),1501-1505.
[4]Huijsmans, C.B, De Pagter, B. Intertible disjointness preserving operators[J]. Proc. of the Edinburgh Mat. Soc.37 (1993),125-132.
[5]Y. A. Abramovich, A. K. Kitover. Inverses of Disjointness Preserving Operators[J]. Memoirs of the Amer. Math. Soc.,143(2000), No.679.
[6]Turan, B. On Ideal Operators[J]. Positivity, (2003),7(1-2),141-148.
[7]Meyer-Nieberg, P. Banach lattices[M]. Springer-Verlag, Berlin, Heideberg, New York,1991.
[8]Y. A. Abramovich and A. K. Kitover. A characterization of operators preserving disjointness in terms of their inverse[J]. Positivity,4(2000),205-212.
[9]C. B. Huijsmans, W. A. J. Luxemburg. Positive Operators and Semigroups on Banach Lattices[J]. Kluwer, Dordrecht,1992. MR 93e:47003.
[10]Y. A. Abramovich. Multiplicative representation of disjointness preserving operators [J]. Netherl. Acad. Wetensch. Proc. Ser. A 86 (1983),265-279. M85f:47040.
[11]Ayse Uyar. Solutions of Two Problems in the Theory of Disjointness Preserving Operators[J]. Positivity 11 (2007),119-121.
[12]Y. A. Abramovich, A. K. Kitover. d-independence and d-bases in vector lattices[J]. Positivity 7 (2003),95-97.
[13]C.B.Huijsmans, B.Depagter, Disjointness preserving and diffuse operator[J]. Compsitio Mathematiea,1991, vol79:351-374.
[14]艾富菊.经典Banach格上保不交算子的性质[D].西南交通大学,2008
[15]C.D.Aliprantis,O.Burkin shaw,Positive Operators[M],Academic Press,INC.,New-York London,1985.
[16]Fethi Benamor,Karim Boulabiar. Maximal ideals of disjointness preserving operators,[J].J Mathematical analysis and applications,2006, Vol322.
[13]C.B.Huijsmans,A.W.Wickstead. The inverse of band preserving and disjointness preserving operators,Indag[J].Mathem.N.S.1992,,vols
[17]Yuri, A. Abramovich, Arkady K.kitover, A characterization of operators preserving disjointness in terms of their inverse [J],Posotovoty,2000,vol 4
[18]陈滋利.Banach格上正则算子格[J].数学学报,2000,(02)
[19]Xiao-dong Zhang,Decomposition theorems for disjointness preserving operators[J],J.Functional Analysis,1993,vol 116.
[20]C.B.Huijsmans, some remarks on disjointness preserving operators [J], Acta Appl.Mathe.,1992.vol 7
[21]W.A.J.Luxemberg,A.C.Zaanen,Riesz Space I [M],North-Holland, Amsterdam New York-Oxford 1971
[23]冯颖,陈滋利,冯天祥,一类经典Banach格上保不交算子的矩阵刻画[J].西南交通大学学报,2003,38卷(4)
[24]A. V. Koldunov, Hammerstein operators preserving disjointness [J], Proceeding of AMS,1995,vol 123
[25]B.De Pagter,A.R.Schep,Band decompositions for disjointness pesering operators [J]Positivaty,2000,vol4
[26]陈滋利.《A Characterization of Non-regular operaters》数学进展,2000,(10)
[27]Adriaan C Zaanen. Introduction to Operator Theory in Riesz Spaces.Berlin-Heidelberg-New York:Springer Verlag,1996.
[28]Koldunov,A. Hammerstein operators preserving disjointness.Proc. Amer. Math. Soc. 1995,123.
[29]Abramovich,Y.A.,Veksler,A.I.,Koldunov,A.V. On operators preserving disjointness.Soviet Math. Dokl.1979.
[30]B.de Pagter. A note on disjointness preserving operator [J]. Proc.Amer.Math.soc. 90(1984).
[31]A.K.Kitover. On disjoint operators in Banach Lattices [J]. Soviet.Math.Dokl250, N4,800-803(1980)
[32]Y.A.Abramovich and A.K.Kitover. Bijective disjointness preserving operator[J]. Functional Analysis and Economic Theorey, Springer,Berlin New York,1998, pp.1-8.
[34]P. T. N. McPolin and A. W. Wickstead, The order boundedness of band preserving operators on uniformly complete vector lattices, Math. Proc. Cambridge Phil. Soc.97 (1985),481-487.
[35]Y. A. Abramovich, E. L. Arenson, A. K. Kitover, Banach C (K)-modules and operators preserving disjointness[J]. Pitman Research Notes in Mathematical Series #277, Longman Scientific & Technical,1992.
[36]W.A.J.Luxemburg, Some Aspects of the Theory of Riesz space[J].Univ. Arkansas Lecture Notes in Math.,4,1979.
[37]Y.A.Abramovich and A.K.Kitover. Inverse and regularity of disjointness preserving operators[J]. Institute of Mathematics,2005.
[38]B.Z.Vulikh. On linear multiplicative operators[J]. Dokl.Akad. Nauk USSR 41(1943), 148-151.
[39]B.Z.Vulikh.Multiplicative in linear semi-order spaces and its application to the theory of operations[J]. Mat.Sbomik 22(1948),267-317.
[40]B.Z.Vulikh. Introduction to the theory of partially ordered spsces[M]. Wolters-Noordhoff Sci.Publication, Groningen,1967.
[41]K.Jarosz. Automatic continuity of separating linear isomorphisms[J]. Canad.Math.Bull.33(1990) 139-144.
[42]Y.A.Abramovich, A.I.Veksler, A.V.Koldunov. Operators preserving disjoitness their continuity and multiplicative representation[J]. Linear operators and their appl. SbornikNauchn. Trudov, Leningrad,1981,13-34.
[43]J.Araujo and J.J.Font. Linear isometries between subspaces of continuous functions[J], Trans.Amer.Math.Soc.349(1997),413-428.
[44]J.J.Font.Automatic continuity of certain isomorphisms between regular Banach function algebras [M]. Glasgo Math.J.39,(1997),333-343.
[45]J.J.Font. Disjointness preserving mappings berween Fourier algebras[J].Proceed. Edinburgh.Math.Soc., submitted.
[46]J.J.Font and S.Hernandez. On separating maps between locally compact spaces[J].Archiv der Math.63(1994),158-165.
[47]J.J.Font and S.Hernandez. Automatic continuity and representation of certain linear isomorphisms between group algebras[J]. Indag. Math.6(1995),397-409.
[48]Z.Ercan, A.W.Wickstead. Toward a theory of non-linear orthomorphisms[J]. Functional Analysis and Economic Theory, Springer, Berlin New York,1998, pp65-87.
[49]A.V.koldunov. Order continuity and representation of operators with the Hammerstein property in vector lattices[J]. Bull.Pol.Acad.Mat.41(1993),43-50.
[50]Y.A.Abramovich and A.K.Kitover. Inverses and regularity of band preserving operators[J]. Indag. Mathem., N.S.,13(2),143-167.