Let X and Y be two normed spaces.Let U be a non-principal ultrafilter on N.Let g:X→Y be a standard ε-phase isometry for someε≥ 0,i.e.,g(0)=0,and for all u.v ∈ X,||‖g(u)+g(v)‖±‖g(u)-g(v)‖|-|‖u+v‖±...Let X and Y be two normed spaces.Let U be a non-principal ultrafilter on N.Let g:X→Y be a standard ε-phase isometry for someε≥ 0,i.e.,g(0)=0,and for all u.v ∈ X,||‖g(u)+g(v)‖±‖g(u)-g(v)‖|-|‖u+v‖±‖u-v‖| |≤ε.The mapping g is said to be a phase isometry provided that ε=0.In this paper,we show the following universal inequality of g:for each u^(*) ∈ w^(*)-exp ‖u^(*)‖B_(x^(*)),there exist a phase function σ_(u^(*)):X→{-1,1} and φ ∈ Y^(*) with ‖φ‖=‖u^(*)‖≡α satisfying that|(u^(*),u)-σ_(u^(*))(u)<φ,g(u)>)|≤5/2εα,for all u ∈ X.In particular,let X be a smooth Banach space.Then we show the following:(1) the universal inequality holds for all u^(*) ∈ X^(*);(2) the constant 5/2 can be reduced to 3/2 provided that Y~*is strictly convex;(3) the existence of such a g implies the existence of a phase isometryΘ:X→Y such that■ provided that Y^(**) has the w^(*)-Kadec-Klee property(for example,Y is both reflexive and locally uniformly convex).展开更多
Let μ,ν be measures with diem L^1(μ) =dim L~∞(ν)=∞ and ⊿ be a 'subinterval' of the real line. Let E=L~∞(ν) or C_b(⊿), in this paper it turns out that the IAP for the space B(L^1(μ)→E) has a negativ...Let μ,ν be measures with diem L^1(μ) =dim L~∞(ν)=∞ and ⊿ be a 'subinterval' of the real line. Let E=L~∞(ν) or C_b(⊿), in this paper it turns out that the IAP for the space B(L^1(μ)→E) has a negative answer.展开更多
Boolean homomorphisms of a hypercube, which correspond to the morphisms in the category of finite Boolean algebras, coincide with the linear isometries of the category of finite binary metric vector spaces.
In this article we bounded symmetric domains study holomorphic isometries of the Poincare disk into Earlier we solved the problem of analytic continuation of germs of holomorphic maps between bounded domains which a...In this article we bounded symmetric domains study holomorphic isometries of the Poincare disk into Earlier we solved the problem of analytic continuation of germs of holomorphic maps between bounded domains which are isometrics up to normalizing constants with respect to the Bergman metric, showing in particular that the graph 170 of any germ of holomorphic isometry of the Poincar6 disk A into an irreducible bounded symmetric domain Ω belong to C^N in its Harish-Chandra realization must extend to an affinealgebraic subvariety V belong to C × C^N = C^N+1, and that the irreducible component of V ∩ (△ × Ω) containing V0 is the graph of a proper holomorphic isometric embedding F : A→ Ω. In this article we study holomorphie isometric embeddings which are asymptotically geodesic at a general boundary point b ∈ δ△. Starting with the structural equation for holomorphic isometrics arising from the Gauss equation, we obtain by covariant differentiation an identity relating certain holomorphic bisectional curvatures to the boundary behavior of the second fundamental form σ of the holomorphie isometric embedding. Using the nonpositivity of holomorphic bisectional curvatures on a bounded symmetric domain, we prove that ‖σ‖ must vanish at a general boundary point either to the order 1 or to the order 1/2, called a holomorphie isometry of the first resp. second kind. We deal with special cases of non-standard holomorphic isometric embeddings of such maps, showing that they must be asymptotically totally geodesic at a general boundary point and in fact of the first kind whenever the target domain is a Cartesian product of complex unit balls. We also study the boundary behavior of an example of holomorphic isometric embedding from the Poincare disk into a Siegel upper half-plane by an explicit determination of the boundary behavior of holomorphic sectional curvatures in the directions tangent to the embedded Poincare disk, showing that the map is indeed asymptotically totally geodesic at a general boundary point and of the first kind. For the metric computation we make use of formulas for symplectic geometry on Siegel upper half-planes.展开更多
Assume that X and Y are real Banach spaces with the same finite dimension.In this paper we show that if a standard coarse isometry f:X→Y satisfies an integral convergence condition or weak stability on a basis,then t...Assume that X and Y are real Banach spaces with the same finite dimension.In this paper we show that if a standard coarse isometry f:X→Y satisfies an integral convergence condition or weak stability on a basis,then there exists a surjective linear isometry U:X→Y such that∥f(x)−Ux∥=o(∥x∥)as∥x∥→∞.This is a generalization about the result of Lindenstrauss and Szankowski on the same finite dimensional Banach spaces without the assumption of surjectivity.As a consequence,we also obtain a stability result forε-isometries which was established by Dilworth.展开更多
In this paper,we discuss the invariant measures for planar piecewise isometries.It is shown that the Hausdorff measure restricted to an almost invariant set with respect to the Hausdorff measure is invariant.
Let X and Y be real Banach spaces. The stability of Hyers Ulam Rassias approximate isometries on restricted domains S (unbounded or bounded) for into mapping f: S→Y satisfying ‖ f(x)-f(y)‖-‖x-y‖≤ε(x,y) for al...Let X and Y be real Banach spaces. The stability of Hyers Ulam Rassias approximate isometries on restricted domains S (unbounded or bounded) for into mapping f: S→Y satisfying ‖ f(x)-f(y)‖-‖x-y‖≤ε(x,y) for all x,y∈S is studied in case that the target space Y is uniformly convex Banach space of the modulus of convexity of power type q ≥2 or Y is the L q(Ω,,μ) (1<q <+∞) space or Y is a Hilbert space. Furthermore, the stability of approximate isometries for the case that (x,y)=‖x‖ p+‖y‖ p or (x,y)=‖x-y‖ p for p ≠1 is investigated.展开更多
In this paper, we give four general results on linear extension of isometries between the unit spheres in β-normed spaces. These results improve the corresponding theorems in β-normed spaces.
In this paper we prove that isometries with respect to the Kobayashi metric between certain domains having boundary points at which the boundary is infinitely flat extend continuously to the boundary.The strategy is t...In this paper we prove that isometries with respect to the Kobayashi metric between certain domains having boundary points at which the boundary is infinitely flat extend continuously to the boundary.The strategy is to reestablish the Gehring-Hayman-type Theorem for these complex domains.Furthermore,the regularity of boundary extension map is given.展开更多
We propose a Specht triangle discretization for a geometrically nonlinear Kirchhoff plate model with large bending isometry.A combination of an adaptive time-stepping gradient flow and a Newton’s method is employed t...We propose a Specht triangle discretization for a geometrically nonlinear Kirchhoff plate model with large bending isometry.A combination of an adaptive time-stepping gradient flow and a Newton’s method is employed to solve the ensuing nonlinear minimization problem.Γ−convergence of the Specht triangle discretization and the unconditional stability of the gradient flow algorithm are proved.We present several numerical examples to demonstrate that our approach significantly enhances both the computational efficiency and accuracy.展开更多
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. A...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.展开更多
A number of previous papers have studied the problem of recovering low-rank matrices with noise, further combining the noisy and perturbed cases, we propose a nonconvex Schatten p-norm minimization method to deal with...A number of previous papers have studied the problem of recovering low-rank matrices with noise, further combining the noisy and perturbed cases, we propose a nonconvex Schatten p-norm minimization method to deal with the recovery of fully perturbed low-rank matrices. By utilizing the p-null space property (p-NSP) and the p-restricted isometry property (p-RIP) of the matrix, sufficient conditions to ensure that the stable and accurate reconstruction for low-rank matrix in the case of full perturbation are derived, and two upper bound recovery error estimation ns are given. These estimations are characterized by two vital aspects, one involving the best r-approximation error and the other concerning the overall noise. Specifically, this paper obtains two new error upper bounds based on the fact that p-RIP and p-NSP are able to recover accurately and stably low-rank matrix, and to some extent improve the conditions corresponding to RIP.展开更多
In this paper, we study the extension of isometries between the unit spheresof some Banach spaces E and the spaces C(Ω). We obtain that if the set sm.S_1(E) of all smoothpoints of the unit sphere S_1(E) is dense in S...In this paper, we study the extension of isometries between the unit spheresof some Banach spaces E and the spaces C(Ω). We obtain that if the set sm.S_1(E) of all smoothpoints of the unit sphere S_1(E) is dense in S_1(E), then under some condition, every surjectiveisometry V_0 from S_1(E) onto S_1(C(Ω)) can be extended to be a real linearly isometric map V of Eonto C(Ω). From this result we also obtain some corollaries. This is the first time we study thisproblem on different typical spaces, and the method of proof is also very different too.展开更多
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)...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.展开更多
An(F)-space E is said to be locally midpoint constricted (in short, Imp-constricted) if there exists some δ】0 such that D(A/2)【D(A) for every subset A of E with 0【D(A)≤δ, where D(A) denotes the diameter of A. Ou...An(F)-space E is said to be locally midpoint constricted (in short, Imp-constricted) if there exists some δ】0 such that D(A/2)【D(A) for every subset A of E with 0【D(A)≤δ, where D(A) denotes the diameter of A. Our main result goes as follow: Let E be an Imp-constricted (F)-space and U an open connected subset of E. Assume that T:U→F is an isometry (i.e., a distance-preserving map) which maps U onto an open subset of the (F)-space F. Then T can be extended to an affine homeomorphism from E to F. Also, some other results about the question whether each isometry between two (F)-spaces is affine are obtained.展开更多
In this paper we study the isometric extension problem and show that every surjective isometry between the unit spheres of L^p(μ) (1 〈 p 〈∞, p ≠ 2) and a Banach space E can be extended to a linear isometry fr...In this paper we study the isometric extension problem and show that every surjective isometry between the unit spheres of L^p(μ) (1 〈 p 〈∞, p ≠ 2) and a Banach space E can be extended to a linear isometry from L^p(μ) onto E, which means that if the unit sphere of E is (metrically) isometric to the unit sphere of L^P(μ), then E is linearly isometric to L^p(μ). We also prove that every surjective 1-Lipschitz or anti-l-Lipschitz map between the unit spheres of L^p(μ1, H1) and L^p(μ2, H2) must be an isometry and can be extended to a linear isometry from L^p(μ2, H2) to L^p(μ2, H2), where H1 and H2 are Hilbert spaces.展开更多
Suppose that f:Hn → Hn (n≥2) maps any r-dimensional hyperplane (1≤r<n) into an r-dimensional hyperplane. In this paper, we prove that f is an isometry if and only if f is a surjective map. This result gives an a...Suppose that f:Hn → Hn (n≥2) maps any r-dimensional hyperplane (1≤r<n) into an r-dimensional hyperplane. In this paper, we prove that f is an isometry if and only if f is a surjective map. This result gives an affirmative answer to a recent conjecture due to Li and Yao.展开更多
Motivated by problems arising from Arithmetic Geometry,in an earlier article one of the authors studied germs of holomorphic isometries between bounded domains with respect to the Bergman metric.In the case of a germ ...Motivated by problems arising from Arithmetic Geometry,in an earlier article one of the authors studied germs of holomorphic isometries between bounded domains with respect to the Bergman metric.In the case of a germ of holomorphic isometry f:(Δ,λ ds 2Δ ;0) → (Ω,ds 2Ω ;0) of the Poincar disk Δ into a bounded symmetric domain Ω C N in its Harish-Chandra realization and equipped with the Bergman metric,f extends to a proper holomorphic isometric embedding F:(Δ,λ ds 2Δ) → (Ω,ds 2Ω) and Graph(f) extends to an affine-algebraic variety V C × C N.Examples of F which are not totally geodesic have been constructed.They arise primarily from the p-th root map ρ p:H → H p and a non-standard holomorphic embedding G from the upper half-plane to the Siegel upper half-plane H 3 of genus 3.In the current article on the one hand we examine second fundamental forms σ of these known examples,by computing explicitly σ 2.On the other hand we study on the theoretical side asymptotic properties of σ for arbitrary holomorphic isometries of the Poincar disk into polydisks.For such mappings expressing via the inverse Cayley transform in terms of the Euclidean coordinate τ=s + it on the upper half-plane H,we have φ(τ)=t 2 u(τ),where u t=0 ≡ 0.We show that u must satisfy the first order differential equation u t | t=0 ≡ 0 on the real axis outside a finite number of points at which u is singular.As a by-product of our method of proof we show that any non-standard holomorphic isometric embedding of the Poincar disk into the polydisk must develop singularities along the boundary circle.The equation φuφt | t=0 ≡ 0 along the real axis for holomorphic isometries into polydisks distinguishes the latter maps from holomorphic isometries into Siegel upper half-planes arising from G.Towards the end of the article we formulate characterization problems for holomorphic isometries suggested both by the theoretical and the computational results of the article.展开更多
for a proper subdomain D of R^(n) and for all x,y∈D defineμD(x,y)=infC_(xy)Cap(D,C_(xy)),where the infimum is taken over all curves Cxy=γ[0,1]in D withγ(0)=x andγ(1)=y,and Cap denotes the conformal capacity of co...for a proper subdomain D of R^(n) and for all x,y∈D defineμD(x,y)=infC_(xy)Cap(D,C_(xy)),where the infimum is taken over all curves Cxy=γ[0,1]in D withγ(0)=x andγ(1)=y,and Cap denotes the conformal capacity of condensers.The quantityμD is a metric if and only if the domain D has a boundary of positive conformal capacity.If Cap(∂D)>0,we callμD the modulus metric of D.Ferrand et al.(1991)have conjectured that isometries for the modulus metric are conformal mappings.Very recently,this conjecture has been proved for n=2 by Betsakos and Pouliasis(2019).In this paper,we prove that the conjecture is also true in higher dimensions n⩾3.展开更多
In this paper, we will discuss some properties of biprojection-commutative elements which are relevant to the classification of certain infinite C*-algebras,, and define an important invariant s(A) of C*-algebra A...In this paper, we will discuss some properties of biprojection-commutative elements which are relevant to the classification of certain infinite C*-algebras,, and define an important invariant s(A) of C*-algebra A as well as give some basic properties with regard to s(A). Moreover we prove that the invariant s(A) has continuity.展开更多
基金supported by the NSFC(12126329,12171266,12126346)the NSF of Fujian Province of China(2023J01805)+5 种基金the Research Start-Up Fund of Jimei University(ZQ2021017)supported by the NSFC(12101234)the NSF of Hebei Province(A2022502010)the Fundamental Research Funds for the Central Universities(2023MS164)the China Scholarship Councilsupported by the Simons Foundation(585081)。
文摘Let X and Y be two normed spaces.Let U be a non-principal ultrafilter on N.Let g:X→Y be a standard ε-phase isometry for someε≥ 0,i.e.,g(0)=0,and for all u.v ∈ X,||‖g(u)+g(v)‖±‖g(u)-g(v)‖|-|‖u+v‖±‖u-v‖| |≤ε.The mapping g is said to be a phase isometry provided that ε=0.In this paper,we show the following universal inequality of g:for each u^(*) ∈ w^(*)-exp ‖u^(*)‖B_(x^(*)),there exist a phase function σ_(u^(*)):X→{-1,1} and φ ∈ Y^(*) with ‖φ‖=‖u^(*)‖≡α satisfying that|(u^(*),u)-σ_(u^(*))(u)<φ,g(u)>)|≤5/2εα,for all u ∈ X.In particular,let X be a smooth Banach space.Then we show the following:(1) the universal inequality holds for all u^(*) ∈ X^(*);(2) the constant 5/2 can be reduced to 3/2 provided that Y~*is strictly convex;(3) the existence of such a g implies the existence of a phase isometryΘ:X→Y such that■ provided that Y^(**) has the w^(*)-Kadec-Klee property(for example,Y is both reflexive and locally uniformly convex).
文摘Let μ,ν be measures with diem L^1(μ) =dim L~∞(ν)=∞ and ⊿ be a 'subinterval' of the real line. Let E=L~∞(ν) or C_b(⊿), in this paper it turns out that the IAP for the space B(L^1(μ)→E) has a negative answer.
文摘Boolean homomorphisms of a hypercube, which correspond to the morphisms in the category of finite Boolean algebras, coincide with the linear isometries of the category of finite binary metric vector spaces.
基金supported by the CERG grant HKU701803 of the Research Grants Council, Hong Kong
文摘In this article we bounded symmetric domains study holomorphic isometries of the Poincare disk into Earlier we solved the problem of analytic continuation of germs of holomorphic maps between bounded domains which are isometrics up to normalizing constants with respect to the Bergman metric, showing in particular that the graph 170 of any germ of holomorphic isometry of the Poincar6 disk A into an irreducible bounded symmetric domain Ω belong to C^N in its Harish-Chandra realization must extend to an affinealgebraic subvariety V belong to C × C^N = C^N+1, and that the irreducible component of V ∩ (△ × Ω) containing V0 is the graph of a proper holomorphic isometric embedding F : A→ Ω. In this article we study holomorphie isometric embeddings which are asymptotically geodesic at a general boundary point b ∈ δ△. Starting with the structural equation for holomorphic isometrics arising from the Gauss equation, we obtain by covariant differentiation an identity relating certain holomorphic bisectional curvatures to the boundary behavior of the second fundamental form σ of the holomorphie isometric embedding. Using the nonpositivity of holomorphic bisectional curvatures on a bounded symmetric domain, we prove that ‖σ‖ must vanish at a general boundary point either to the order 1 or to the order 1/2, called a holomorphie isometry of the first resp. second kind. We deal with special cases of non-standard holomorphic isometric embeddings of such maps, showing that they must be asymptotically totally geodesic at a general boundary point and in fact of the first kind whenever the target domain is a Cartesian product of complex unit balls. We also study the boundary behavior of an example of holomorphic isometric embedding from the Poincare disk into a Siegel upper half-plane by an explicit determination of the boundary behavior of holomorphic sectional curvatures in the directions tangent to the embedded Poincare disk, showing that the map is indeed asymptotically totally geodesic at a general boundary point and of the first kind. For the metric computation we make use of formulas for symplectic geometry on Siegel upper half-planes.
基金Supported by National Natural Science Foundation of China(11731010 and 12071388)。
文摘Assume that X and Y are real Banach spaces with the same finite dimension.In this paper we show that if a standard coarse isometry f:X→Y satisfies an integral convergence condition or weak stability on a basis,then there exists a surjective linear isometry U:X→Y such that∥f(x)−Ux∥=o(∥x∥)as∥x∥→∞.This is a generalization about the result of Lindenstrauss and Szankowski on the same finite dimensional Banach spaces without the assumption of surjectivity.As a consequence,we also obtain a stability result forε-isometries which was established by Dilworth.
基金Project supported by the National Natural Science Foundation of China (Grant No.10672146)the Natural Science Foundation of Jiangxi Province (Grant No.2007GQS0142)
文摘In this paper,we discuss the invariant measures for planar piecewise isometries.It is shown that the Hausdorff measure restricted to an almost invariant set with respect to the Hausdorff measure is invariant.
文摘Let X and Y be real Banach spaces. The stability of Hyers Ulam Rassias approximate isometries on restricted domains S (unbounded or bounded) for into mapping f: S→Y satisfying ‖ f(x)-f(y)‖-‖x-y‖≤ε(x,y) for all x,y∈S is studied in case that the target space Y is uniformly convex Banach space of the modulus of convexity of power type q ≥2 or Y is the L q(Ω,,μ) (1<q <+∞) space or Y is a Hilbert space. Furthermore, the stability of approximate isometries for the case that (x,y)=‖x‖ p+‖y‖ p or (x,y)=‖x-y‖ p for p ≠1 is investigated.
文摘In this paper, we give four general results on linear extension of isometries between the unit spheres in β-normed spaces. These results improve the corresponding theorems in β-normed spaces.
基金supported by National Key R&D Program of China(Grant No.2021YFA1003100)NSFC(Grant Nos.11925107 and 12226334)+2 种基金Key Research Program of Frontier Sciences,CAS(Grant No.ZDBS-LY-7002)supported by the Young Scientist Program of the Ministry of Science and Technology of China(Grant No.2021YFA1002200)NSFC(Grant No.12201059)。
文摘In this paper we prove that isometries with respect to the Kobayashi metric between certain domains having boundary points at which the boundary is infinitely flat extend continuously to the boundary.The strategy is to reestablish the Gehring-Hayman-type Theorem for these complex domains.Furthermore,the regularity of boundary extension map is given.
基金supported by National Natural Science Foundation of China through Grants No.11971467 and No.12371438.
文摘We propose a Specht triangle discretization for a geometrically nonlinear Kirchhoff plate model with large bending isometry.A combination of an adaptive time-stepping gradient flow and a Newton’s method is employed to solve the ensuing nonlinear minimization problem.Γ−convergence of the Specht triangle discretization and the unconditional stability of the gradient flow algorithm are proved.We present several numerical examples to demonstrate that our approach significantly enhances both the computational efficiency and accuracy.
基金supported by the Spanish Ministry of Economy and Competitiveness and European Regional Development Fund (Grant No. MTM2014-58984-P)Junta de Andalucía (Grant No. FQM375)+1 种基金Grants-in-Aid for Scientific Research (Grant No. 16J01162)Japan Society for the Promotion of Science
文摘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.
文摘A number of previous papers have studied the problem of recovering low-rank matrices with noise, further combining the noisy and perturbed cases, we propose a nonconvex Schatten p-norm minimization method to deal with the recovery of fully perturbed low-rank matrices. By utilizing the p-null space property (p-NSP) and the p-restricted isometry property (p-RIP) of the matrix, sufficient conditions to ensure that the stable and accurate reconstruction for low-rank matrix in the case of full perturbation are derived, and two upper bound recovery error estimation ns are given. These estimations are characterized by two vital aspects, one involving the best r-approximation error and the other concerning the overall noise. Specifically, this paper obtains two new error upper bounds based on the fact that p-RIP and p-NSP are able to recover accurately and stably low-rank matrix, and to some extent improve the conditions corresponding to RIP.
文摘In this paper, we study the extension of isometries between the unit spheresof some Banach spaces E and the spaces C(Ω). We obtain that if the set sm.S_1(E) of all smoothpoints of the unit sphere S_1(E) is dense in S_1(E), then under some condition, every surjectiveisometry V_0 from S_1(E) onto S_1(C(Ω)) can be extended to be a real linearly isometric map V of Eonto C(Ω). From this result we also obtain some corollaries. This is the first time we study thisproblem on different typical spaces, and the method of proof is also very different too.
基金supported by Natural Science Foundation of the Education Departmentof Jiangsu Province (Grant No.09KJD110003)
文摘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.
文摘An(F)-space E is said to be locally midpoint constricted (in short, Imp-constricted) if there exists some δ】0 such that D(A/2)【D(A) for every subset A of E with 0【D(A)≤δ, where D(A) denotes the diameter of A. Our main result goes as follow: Let E be an Imp-constricted (F)-space and U an open connected subset of E. Assume that T:U→F is an isometry (i.e., a distance-preserving map) which maps U onto an open subset of the (F)-space F. Then T can be extended to an affine homeomorphism from E to F. Also, some other results about the question whether each isometry between two (F)-spaces is affine are obtained.
基金Supported by the Fundamental Research Funds for the Central UniversitiesNational Natural Science Foundation of China (Grant No. 10871101)
文摘In this paper we study the isometric extension problem and show that every surjective isometry between the unit spheres of L^p(μ) (1 〈 p 〈∞, p ≠ 2) and a Banach space E can be extended to a linear isometry from L^p(μ) onto E, which means that if the unit sphere of E is (metrically) isometric to the unit sphere of L^P(μ), then E is linearly isometric to L^p(μ). We also prove that every surjective 1-Lipschitz or anti-l-Lipschitz map between the unit spheres of L^p(μ1, H1) and L^p(μ2, H2) must be an isometry and can be extended to a linear isometry from L^p(μ2, H2) to L^p(μ2, H2), where H1 and H2 are Hilbert spaces.
基金supported by National Natural Science Foundation of China (Grant No.10771059)Tianyuan Foundation
文摘Suppose that f:Hn → Hn (n≥2) maps any r-dimensional hyperplane (1≤r<n) into an r-dimensional hyperplane. In this paper, we prove that f is an isometry if and only if f is a surjective map. This result gives an affirmative answer to a recent conjecture due to Li and Yao.
基金supported by the Research Grants Council of Hong Kong,China (Grant No. CERG 7018/03)
文摘Motivated by problems arising from Arithmetic Geometry,in an earlier article one of the authors studied germs of holomorphic isometries between bounded domains with respect to the Bergman metric.In the case of a germ of holomorphic isometry f:(Δ,λ ds 2Δ ;0) → (Ω,ds 2Ω ;0) of the Poincar disk Δ into a bounded symmetric domain Ω C N in its Harish-Chandra realization and equipped with the Bergman metric,f extends to a proper holomorphic isometric embedding F:(Δ,λ ds 2Δ) → (Ω,ds 2Ω) and Graph(f) extends to an affine-algebraic variety V C × C N.Examples of F which are not totally geodesic have been constructed.They arise primarily from the p-th root map ρ p:H → H p and a non-standard holomorphic embedding G from the upper half-plane to the Siegel upper half-plane H 3 of genus 3.In the current article on the one hand we examine second fundamental forms σ of these known examples,by computing explicitly σ 2.On the other hand we study on the theoretical side asymptotic properties of σ for arbitrary holomorphic isometries of the Poincar disk into polydisks.For such mappings expressing via the inverse Cayley transform in terms of the Euclidean coordinate τ=s + it on the upper half-plane H,we have φ(τ)=t 2 u(τ),where u t=0 ≡ 0.We show that u must satisfy the first order differential equation u t | t=0 ≡ 0 on the real axis outside a finite number of points at which u is singular.As a by-product of our method of proof we show that any non-standard holomorphic isometric embedding of the Poincar disk into the polydisk must develop singularities along the boundary circle.The equation φuφt | t=0 ≡ 0 along the real axis for holomorphic isometries into polydisks distinguishes the latter maps from holomorphic isometries into Siegel upper half-planes arising from G.Towards the end of the article we formulate characterization problems for holomorphic isometries suggested both by the theoretical and the computational results of the article.
基金supported by National Natural Science Foundation of China(Grant Nos.11771400 and 11911530457)Science Foundation of Zhejiang Sci-Tech University(Grant No.16062023Y)。
文摘for a proper subdomain D of R^(n) and for all x,y∈D defineμD(x,y)=infC_(xy)Cap(D,C_(xy)),where the infimum is taken over all curves Cxy=γ[0,1]in D withγ(0)=x andγ(1)=y,and Cap denotes the conformal capacity of condensers.The quantityμD is a metric if and only if the domain D has a boundary of positive conformal capacity.If Cap(∂D)>0,we callμD the modulus metric of D.Ferrand et al.(1991)have conjectured that isometries for the modulus metric are conformal mappings.Very recently,this conjecture has been proved for n=2 by Betsakos and Pouliasis(2019).In this paper,we prove that the conjecture is also true in higher dimensions n⩾3.
基金Supported by National Natural Science Foundation of China (Grant No. 10771161)
文摘In this paper, we will discuss some properties of biprojection-commutative elements which are relevant to the classification of certain infinite C*-algebras,, and define an important invariant s(A) of C*-algebra A as well as give some basic properties with regard to s(A). Moreover we prove that the invariant s(A) has continuity.