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
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
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 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.
