On Extension of Isometries Between the Unit Spheres of Normed Space E and lP（p 〉 1）

In this paper, we study the extension of isometries between the unit spheres of normed space E and lP（p 〉 1）. We arrive at a conclusion that any surjective isometry between the unit sphere of normed space lP（p 〉 1） and E can be extended to be a linear isometry on the whole space lP（p 〉 1） under some condition.     

EXTENSION OF ISOMETRIES BETWEEN THE UNIT SPHERES OF COMPLEX lp(Γ)(p > 1) SPACES

In this paper, we study the extension of isometries between the unit spheres of complex Banach spaces lp(Γ) and lp(△)(p 1). We first derive the representation of isometries between the unit spheres of complex Banach spaces lp(Γ) and lp(△). Then we arrive at a conclusion that any surjective isometry between the unit spheres of complex Banach spaces lp(Γ)and lp(△) can be extended to be a linear isometry on the whole space.     

Extension of isometries on the unit sphere of L p spaces

In this paper we study the isometric extension problem and show that every surjective isometry between the unit spheres of Lp (μ) (1 p ∞, p≠2) and a Banach space E can be extended to a linear isometry from Lp(μ) onto E, which means that if the unit sphere of E is (metrically) isometric to the unit sphere of Lp(μ), then E is linearly isometric to Lp(μ). We also prove that every surjective 1-Lipschitz or anti-1-Lipschitz map between the unit spheres of Lp (μ1, H1) and Lp(μ2,H2) must be an isometry and can be extended to a linear isometry from Lp (μ1,H1) to Lp (μ2,H2), where H1 and H2 are Hilbert spaces.     

THE ISOMETRIC EXTENSION OF AN INTO MAPPING FROM THE UNIT SPHERE S[e（Γ）] TO THE UNIT SPHERE S（E）

This is such a article to consider an ＂into＂ isometric mapping between two unit spheres of two infinite dimensional spaces of different types. In this article, we find a useful condition （using the Krein-Milman property） under which an into-isometric mapping from the unit sphere of e（Γ） to the unit sphere of a normed space E can be linearly isometric extended.     

On the Extension of Isometries between Unit Spheres of E and C（Ω）

In this paper,we study the extension of isometries between the unit spheres of some Banach spaces E and the spaces C(Ω). We obtain that if the set sm.S1(E) of all smooth points of the unit sphere S1(E) is dense in S1(E),then under some condition,every surjective isometry V0 from S1(E) onto S1(C(Ω)) can be extended to be a real linearly isometric map V of E onto C(Ω).From this resultwe also obtain some corollaries. This is the first time we study this problem on different typical spaces,and the method of proof is also very different too.     

On Linearly Isometric Extensions for 1-Lipschitz Mappings Between Unit Spheres of ALP-spaces （p 〉 2）

In this paper, we show that if Vo is a 1-Lipschitz mapping between unit spheres of two ALP-spaces with p 〉 2 and -Vo（S1（LP）） C Vo（S1（LP））, then V0 can be extended to a linear isometry defined on the whole space. If 1 〈 p 〈 2 and Vo is an ＂anti-l-Lipschitz＂ mapping, then Vo can also be linearly and isometrically extended.     

Isometries on the Quasi-Banach Spaces Lp （0〈p〈1）

We study the extension of isometries between the unit spheres of quasi-Banach spaces Lp for 0〈p〈1. We give some sufficient conditions such that an isometric mapping from the the unit sphere of Lp（μ） into that of another LP（ν） can be extended to be a linear isometry defined on the whole space.     

A solution to Tingley’s problem for isometries between the unit spheres of compact C*-algebras and JB*-triples

Let f : S(E) → S(B) be a surjective isometry between the unit spheres of two weakly compact JB*-triples not containing direct summands of rank less than or equal to 3. Suppose E has rank greater than or equal to 5. Applying techniques developed in JB*-triple theory, we prove that f admits an extension to a surjective real linear isometry T : E → B. Among the consequences, we show that every surjective isometry between the unit spheres of two compact C*-algebras A and B, without assuming any restriction on the rank of their direct summands(and in particular when A = K(H) and B = K(H′)), extends to a surjective real linear isometry from A into B. These results provide new examples of infinite-dimensional Banach spaces where Tingley's problem admits a positive answer.     

ON LINEAR EXTENSION OF 1-LIPSCHITZ MAPPING FROM HILBERT SPACE INTO A NORMED SPACE

In this article, we prove that an into 1-Lipschitz mapping from the unit sphere of a Hilbert space to the unit sphere of an arbitrary normed space, which under some conditions, can be extended to be a linear isometry on the whole space.     

像空间为$l^1$的单位球面上的等距延拓问题

The main result of this paper is to prove Fang and Wang＇s result by another method： Let E be any normed linear space and Vo ： S（E）→ S（l^1） be a surjective isometry. Then V0 can be linearly isometrically extended to E.     

The Representation Theorem of onto Isometric Mappings between Two Unit Spheres of l^1（Γ） Type Spaces and The Application to the Isometric Extension Problem

In this paper, we first derive the representation theorem of onto isometric mappings in the unit spheres of l^1（Γ） type spaces, and then conchlde that such mappings can be extended to the whole space as real linear isometries by using a previous result of the author.     

The isometric extension of “into” mappings on unit spheres of AL-spaces

In this paper, we show that if V0 is an isometric mapping from the unit sphere of an AL-space onto the unit sphere of a Banach space E, then V0 can be extended to a linear isometry defined on the whole space.     

Extension of isometries on the unit sphere of l~p (Γ) space

In this paper,we obtain that every isometry from the unit sphere S(l p (Γ)) of l p (Γ) (1 p ∞,p≠2) onto the unit sphere S(E) of a Banach space E can be extended to be a (real) linear isometry of l p (Γ) onto E,so,we give an affirmative answer to the corresponding Tingley's problem.     

ISOMETRIES ON THE SPACE s

In this article, the author presents some results of the isometric linear extension from some spheres in the finite dimensional space S(n). Moreover, the author presents the representation for the onto isometric mappings in the space a. It is obtained that if V is a surjective isometry from the space s onto s with V(0)=0, then V must be real linear.     

ON THE FIGIEL TYPE PROBLEM AND EXTENSION OF ε-ISOMETRY BETWEEN UNIT SPHERES

This paper studies two isometric problems between unit spheres of Banach spaces.In the first part,we introduce and study the Figiel type problem of isometric embeddings between unit spheres.However,the classical Figiel theorem on the whole space cannot be trivially generalized to this case,and this is pointed out by a counterexample.After establishing this,we find a natural necessary condition required by the existence of the Figiel operator.Furthermore,we prove that when X is a space with the T...     

A VIEWPOINT TO MEASURE OF NON-COMPACTNESS OF OPERATORS IN BANACH SPACES

This article is committed to deal with measure of non-compactness of operators in Banach spaces. Firstly, the collection C(X)(consisting of all nonempty closed bounded convex sets of a Banach space X endowed with the uaual set addition and scaler multiplication) is a normed semigroup, and the mapping J from C(X) onto F(?) is a fully order-preserving positively linear surjective isometry, where ? is the closed unit ball of X*and F(?) the collection of all continuous and w*-lower semicontinuous sublinear functions on X*but restricted to ?. Furthermore, both ■ and ■ are Banach lattices and EK is a lattice ideal of EC. The quotient space EC/EK is an abstract M space, hence,order isometric to a sublattice of C(K) for some compact Haudorspace K, and(FQJ)C which is a closed cone is contained in the positive cone of C(K), where Q : E_C → E_C/E_K is the quotient mapping and F : E_C/E_K → C(K) is a corresponding order isometry. Finally, the representation of the measure of non-compactness of operators is given: Let B_X be the closed unit ball of a Banach space X, then■     

The Isometric Extension of an Into Mapping from the Unit Sphere S（t（2）^∞） to S（L^1（μ））

This is the first paper to consider the isometric extension problem of an into-mapping between the unit spheres of two different types of spaces. We prove that, under some conditions, an into-isometric mapping from the unit sphere S（t（2）^∞） to S（L^1（μ） can be （real） linearly isometrically extended.     

$AL_P(\mu,H)$空间中单位球面间1-Lipschitz和反1-Lipschitz映射的线性等距延拓的一个注记

In this paper, we show that if V0 is a 1-Lipschitz mapping between unit spheres of LP (μ, H) and LP(ν,H)(p>2, H is a Hilbert space), and-V0(S(Lp(μ, H )))V0(S(Lp(μ, H))), then V0 can be extended to a linear isometry defined on the whole space. If 1相似文献   

A Characterization of Generalized Monotone Normed Cones

Let C be a cone and consider a quasi-norm p defined on it. We study the structure of the couple （C, p） as a topological space in the case where the function p is also monotone. We characterize when the topology of a quasi-normed cone can be defined by means of a monotone norm. We also define and study the dual cone of a monotone normed cone and the monotone quotient of a general cone. We provide a decomposition theorem which allows us to write a cone as a direct sum of a monotone subcone that is isomorphic to the monotone quotient and other particular subcone.     

A remark on extension of into isometries

In this paper, we prove that an into isometry form S（l（n）^∞） to S（E）, which under some conditions, can be extended to be a linear isometry defined on the whole space. Therefore we improve the results of [Ding, G. G.： The isometric extension of an into mapping from the unit sphere S（l（2）^∞） to S（Lμ^1）. Acta Mathematica Sinica, English Series, 22（6）, 1721-1724 （2006）].