We show that if a bounded linear operator can be approximated by a net(or sequence)of uniformly bounded finite rank Lipschitz mappings pointwisely,then it can be approximated by a net(or sequence)of uniformly bounded ...We show that if a bounded linear operator can be approximated by a net(or sequence)of uniformly bounded finite rank Lipschitz mappings pointwisely,then it can be approximated by a net(or sequence)of uniformly bounded finite rank linear operators under the strong operator topology.As an application,we deduce that a Banach space has an(unconditional)Lipschitz frame if and only if it has an(unconditional)Schauder frame.Another immediate consequence of our result recovers the famous Godefroy-Kalton theorem(Godefroy and Kalton(2003))which says that the Lipschitz bounded approximation property and the bounded approximation property are equivalent for every Banach space.展开更多
A pair of Banach spaces (X, Y) is said to be stable if for every ￡-isometry f : X→Y,there exist γ> 0 and a bounded linear operator T : L(f)→X with ||T||≤α such that ||Tf(x)- x||≤γε for all x ∈ X, where L(...A pair of Banach spaces (X, Y) is said to be stable if for every ￡-isometry f : X→Y,there exist γ> 0 and a bounded linear operator T : L(f)→X with ||T||≤α such that ||Tf(x)- x||≤γε for all x ∈ X, where L(f) is the closed linear span of f(X). In this article, we study the stability of a pair of Banach spaces (X, Y) when X is a C(K) space. This gives a new positive answer to Qian's problem. Finally, we also obtain a nonlinear version for Qian's problem.展开更多
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
基金supported by National Natural Science Foundation of China(Grant Nos.11671214,11971348 and 12071230)Hundred Young Academia Leaders Program of Nankai University(Grant Nos.63223027 and ZB22000105)+2 种基金Undergraduate Education and Teaching Project of Nankai University(Grant No.NKJG2022053)National College Students'Innovation and Entrepreneurship Training Program of Nankai University(Grant No.202210055048)supported by Simons Foundation(Grant No.585081)。
文摘We show that if a bounded linear operator can be approximated by a net(or sequence)of uniformly bounded finite rank Lipschitz mappings pointwisely,then it can be approximated by a net(or sequence)of uniformly bounded finite rank linear operators under the strong operator topology.As an application,we deduce that a Banach space has an(unconditional)Lipschitz frame if and only if it has an(unconditional)Schauder frame.Another immediate consequence of our result recovers the famous Godefroy-Kalton theorem(Godefroy and Kalton(2003))which says that the Lipschitz bounded approximation property and the bounded approximation property are equivalent for every Banach space.
基金supported in part by NSFC(11601264,11471270 and 11471271)the Fundamental Research Funds for the Central Universities(20720160037)+4 种基金the Outstanding Youth Scientific Research Personnel Training Program of Fujian Provincethe High level Talents Innovation and Entrepreneurship Project of Quanzhou City(2017Z032)the Research Foundation of Quanzhou Normal University(2016YYKJ12)the Natural Science Foundation of Fujian Province of China(2019J05103)supported in part by NSFC(11628102)
文摘A pair of Banach spaces (X, Y) is said to be stable if for every ￡-isometry f : X→Y,there exist γ> 0 and a bounded linear operator T : L(f)→X with ||T||≤α such that ||Tf(x)- x||≤γε for all x ∈ X, where L(f) is the closed linear span of f(X). In this article, we study the stability of a pair of Banach spaces (X, Y) when X is a C(K) space. This gives a new positive answer to Qian's problem. Finally, we also obtain a nonlinear version for Qian's problem.
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).
