廣義近似保等分線正交映射

孔　亮,王念良,劉曉民

(商洛學(xué)院 數(shù)學(xué)與計(jì)算機(jī)應(yīng)用學(xué)院 應(yīng)用數(shù)學(xué)研究所,陜西 商洛　726000)

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廣義近似保等分線正交映射

孔亮,王念良,劉曉民

(商洛學(xué)院 數(shù)學(xué)與計(jì)算機(jī)應(yīng)用學(xué)院 應(yīng)用數(shù)學(xué)研究所,陜西 商洛726000)

在實(shí)賦范線性空間中,給出了廣義近似等分線正交的定義和性質(zhì)以及廣義近似保等分線正交映射的定義。運(yùn)用算子論方法,證明了(δ1,δ2)-近似等距是廣義近似保等分線正交映射,得到了有界線性映射成為廣義近似保等分線正交映射的一些充分條件。

近似等距;等分線正交;近似等分線正交;廣義近似保等分線正交映射

正交性是內(nèi)積空間的重要概念。自20世紀(jì)初,許多學(xué)者相繼在賦范線性空間中推廣了內(nèi)積空間中的正交性,得到了各種正交性定義,如R-正交、B-正交、I-正交、P-正交等[1-8],至今新的正交性定義被不斷引入和研究。為了更深入地研究賦范線性空間的幾何性質(zhì),一些學(xué)者在內(nèi)積空間、準(zhǔn)HilbertA-模、賦范線性空間中引入各種近似正交性定義并研究其性質(zhì)[9-12]。同時(shí)保持和近似保持各種正交性映射的性質(zhì),受到了眾多學(xué)者的關(guān)注。文獻(xiàn)[9,10]中在內(nèi)積空間中給出了線性保正交映射的刻畫;文獻(xiàn)[11]中在Hilbert空間中證明了非零近似保正交線性映射有界并且是下有界的;文獻(xiàn)[12]中在準(zhǔn)HilbertA-模中推廣了文獻(xiàn)[11]中的結(jié)果;文獻(xiàn)[13]中在實(shí)賦范線性空間中給出了近似保等分線正交映射的定義和性質(zhì);關(guān)于其他各種近似保正交映射已有許多研究[14-22]。受文獻(xiàn)[1-22]中概念和結(jié)論的啟發(fā),研究在實(shí)賦范線性空間中引入廣義近似保等分線正交映射的定義,研究有界線性映射成為廣義近似保等分線正交映射的一些充分條件。

1　預(yù)備知識

在研究中,X和Y表示實(shí)賦范線性空間,H表示實(shí)Banach空間,表示實(shí)數(shù)域。

定義1設(shè)δ∈[0,1),U∶X→Y是線性映射。若U滿足

(1-δ)‖x‖≤‖U(x)‖≤(1+δ)‖x‖,?x∈X

則稱U是δ-近似等距[14]。

定義2設(shè)δ1,δ2∈[0,1),U∶X→Y是線性映射。若U滿足

(1-δ1)‖x‖≤‖U(x)‖≤(1+δ2)‖x‖,?x∈X

則稱U是(δ1,δ2)-近似等距。

定義4設(shè)x,y∈X,δ∈[0,1)。若

定義5設(shè)x,y∈X,δ1,δ2∈[0,1)。若

命題1設(shè)x,y∈X,δ1,δ2∈[0,1),α,β∈,則:

(1)

若x和y線性相關(guān),則存在s≠0使y=sx,從而由式(1)知

(2)

定義6設(shè)映射T∶X→Y,δ,ε∈[0,1)。若對任意的x,y∈X,xδ⊥Wy?T(x)ε⊥WT(y),則稱T是近似保等分線正交映射[13]。

注2在定義7中,當(dāng)δ1=δ2=δ,ε1=ε2=ε時(shí),T是近似保等分線正交映射,從而定義7是定義6的一個(gè)推廣。

2　主要結(jié)果及其證明

引理1設(shè)x,y∈X{0},則[15]

(1-δ1)‖z‖≤‖U(z)‖≤(1+δ2)‖z‖,?z∈X

(3)

于是

(1-δ1)‖x‖≤‖U(x)‖≤(1+δ2)‖x‖, (1-δ1)‖y‖≤‖U(y)‖≤(1+δ2)‖y‖

和

(4)

(5)

則T是廣義近似保等分線正交映射。

引理2設(shè)δ1,δ2∈[0,1)。若線性映射T∶H→H滿足

‖T(x)-x‖≤δ1‖x‖+δ2‖T(x)‖,?x∈H

則T是有界滿射,且

‖T(x)-x‖≤δ1‖x‖+δ2‖T(x)‖,?x∈H

則T和T-1都是廣義近似保等分線正交映射。

證明由引理2得

(6)

(7)

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Generalized Approximate Preserving Bisectrix Orthogonal Mapping

Kong Liang,Wang Nianliang,Liu Xiaomin

(College of Mathematics and Computer Application,Institute of Applied Mathematics,Shangluo University,Shangluo 726000,China)

In real normal linear space,it gives the definition and property of the generalized approximate bisectrix orthogonality,and also gives the definition of approximate preserving bisectrix orthogonal mapping,and uses the theory of operator to show (δ1,δ2) approximate equidistance is the generalized approximate preserving bisectrix orthogonal map,knowing that the bounded linear mapping is the sufficient condition of generalized approximate preserving bisectrix orthogonal mapping.

Approximate equidistance;Bisectrix orthogonality;Approximate bisectrix orthogonality;Generalized approximate preserving bisectrix orthogonal mapping

10.16468/j.cnki.issn1004-0366.2016.04.004.

2015-09-23;

2015-10-26.

陜西省科技廳科研項(xiàng)目(2014JM1019);陜西省教育廳科研項(xiàng)目(15JK1221);商洛學(xué)院博士團(tuán)隊(duì)服務(wù)地方科技創(chuàng)新與經(jīng)濟(jì)社會發(fā)展能力提升專項(xiàng)子項(xiàng)目(SK2014-01-08);商洛學(xué)院科研項(xiàng)目(14SKY016).

孔亮(1983-),男,陜西商洛人,碩士,講師,研究方向?yàn)榉汉治?E-mail:kongliang2005@163.com.

O177.1

A

1004-0366(2016)04-0013-05

引用格式:Kong Liang,Wang Nianliang,Liu Xiaomin.Generalized Approximate Preserving Bisectrix Orthogonal Mapping[J].Journal of Gansu Sciences,2016,28(4):13-16,22.[孔亮,王念良,劉曉民.廣義近似保等分線正交映射[J].甘肅科學(xué)學(xué)報(bào),2016,28(4):13-16,22.]