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
Based on a characterization of upper semi-Fredholm operators due to A.Lebow and M.Schechter,we introduce and investigate a new quantity characterizing upper semi-Fredholm operators.This quantity and several well-known...Based on a characterization of upper semi-Fredholm operators due to A.Lebow and M.Schechter,we introduce and investigate a new quantity characterizing upper semi-Fredholm operators.This quantity and several well-known quantities are used to characterize bounded compact approximation property.Similarly,a new quantity characterizing lower semi-Fredholm operators is introduced,investigated and used to characterize the bounded compact approximation property for dual spaces.展开更多
In this paper, an equivalent relation among the reflexivity, weak sequential completeness and bounded completeness in full function space is given. Some results on weakly sequential compactness of subset and the prope...In this paper, an equivalent relation among the reflexivity, weak sequential completeness and bounded completeness in full function space is given. Some results on weakly sequential compactness of subset and the property （u） in substitution spaces are obtained.展开更多
A k-colouring （not necessarily proper） of vertices of a graph is called acyclic, if for every pair of distinct colours i and j the subgraph induced by the edges whose endpoints have colours i and j is acyclic. We co...A k-colouring （not necessarily proper） of vertices of a graph is called acyclic, if for every pair of distinct colours i and j the subgraph induced by the edges whose endpoints have colours i and j is acyclic. We consider acyclie k-eolourings such that each colour class induces a graph with a given （hereditary） property. In particular, we consider aeyclic k-eolourings in which each colour class induces a graph with maximum degree at most t, which are referred to as acyclic t-improper k-colourings. The acyelic t-improper chromatic number of a graph G is the smallest k for which there exists an acyclic t-improper k-colouring of G. We focus on acyclic colourings of graphs with maximum degree 4. We prove that 3 is an upper bound for the acyclic 3-improper chromatic number of this class of graphs. We also provide a non-trivial family of graphs with maximum degree 4 whose acyclic 3-improper chromatic number is at most 2, namely, the graphs with maximum average degree at most 3. Finally, we prove that any graph G with A（G） ≤ 4 can be acyelically coloured with 4 colours in such a way that each colour class induces an acyclic graph with maximum degree at most 3.展开更多
In this paper we prove three equivalent conditions of bounded closed convexset K in Banach space to have the drop and weak drop properties. We also give fourequivalent conditions of Banach space and its dual space to ...In this paper we prove three equivalent conditions of bounded closed convexset K in Banach space to have the drop and weak drop properties. We also give fourequivalent conditions of Banach space and its dual space to have the drop and weak dropproperties.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 by the National Natural Science Foundation of China(Grant No.11971403)the Natural Science Foundation of Fujian Province of China(Grant No.2019J01024)。
文摘Based on a characterization of upper semi-Fredholm operators due to A.Lebow and M.Schechter,we introduce and investigate a new quantity characterizing upper semi-Fredholm operators.This quantity and several well-known quantities are used to characterize bounded compact approximation property.Similarly,a new quantity characterizing lower semi-Fredholm operators is introduced,investigated and used to characterize the bounded compact approximation property for dual spaces.
基金Supported by Specific Academic Discipline Project of Shanghai Municipal Education Commission (Grant No. A-3500-11-10)
文摘In this paper, an equivalent relation among the reflexivity, weak sequential completeness and bounded completeness in full function space is given. Some results on weakly sequential compactness of subset and the property （u） in substitution spaces are obtained.
基金supported by the Minister of Science and Higher Education of Poland(Grant No.JP2010009070)
文摘A k-colouring （not necessarily proper） of vertices of a graph is called acyclic, if for every pair of distinct colours i and j the subgraph induced by the edges whose endpoints have colours i and j is acyclic. We consider acyclie k-eolourings such that each colour class induces a graph with a given （hereditary） property. In particular, we consider aeyclic k-eolourings in which each colour class induces a graph with maximum degree at most t, which are referred to as acyclic t-improper k-colourings. The acyelic t-improper chromatic number of a graph G is the smallest k for which there exists an acyclic t-improper k-colouring of G. We focus on acyclic colourings of graphs with maximum degree 4. We prove that 3 is an upper bound for the acyclic 3-improper chromatic number of this class of graphs. We also provide a non-trivial family of graphs with maximum degree 4 whose acyclic 3-improper chromatic number is at most 2, namely, the graphs with maximum average degree at most 3. Finally, we prove that any graph G with A（G） ≤ 4 can be acyelically coloured with 4 colours in such a way that each colour class induces an acyclic graph with maximum degree at most 3.
文摘In this paper we prove three equivalent conditions of bounded closed convexset K in Banach space to have the drop and weak drop properties. We also give fourequivalent conditions of Banach space and its dual space to have the drop and weak dropproperties.
基金Supported by the Nation Natural Science Foundation of China(11472063)the Provincial Natural Science Research Program of Higher Education Institutions of Anhui Province(KJ2013A194,KJ2013Z230)Anhui Province Colleges and Universities Outstanding Youth Talent Support Program(gxyq ZD2016285)
