All-derivable Points In B(h)

Theory of operator algebras produced in the 20th century, 30's, is an extremely important part in the study of functional analysis. It is has extensive contacts and mutual penetration with physics, quantum mechanics, non-commutative geometry, linear systems, control theory, number theory, and other important branch of mathematics. Along with other subjects in the application of this theory has made significant progress, and has become a matter of concern in modern mathematics. In recent years, there has been considerable interest in studying which linear mapping on operator algebras are derivations, this area has been extensively studied by many experts and scholars at home and abroad and has made a lot of scientific research results. In 1990, the notion of local derivations was introduced independently by D. R. Larson and R. V. Kadison. Also in this year , Larson proved that every local derivation on B ( H ) is a derivation, where X is a Banach space. In 1990, Kadison proved that every norm-continuous local derivation on Von Neumann algebra is a derivation. Afterword, Wu Jing, Lu Shijie and Li Pengtong showed that every derivable mappingφat 0 withφ( I) = 0 on nest algebras is an inner derivation.From 2007 to 2009, Zhu Jun and Xiong Changping have proved that (1) every invertible operator on nest algebras is an all-derivable point for the strongly operator topology. (2) ( I is an identity operator) is an all-derivable point of the algebra of all 2×2 operator matrices for the strongly topology. (3) G∈TM2 is an all-derivable point of TM 2 if and only if G≠0 where TM 2 is the algebra of all n×n upper triangular matrices. (4) G∈Un is an all-derivable point in U n if and only if G≠0 where U n is the algebra of all n×n matrices. In 2009, Lu Fangyan has proved on condition that let A is an Banach algebra and X is an left (or right) invertible in A , thenδis a Jordan derivation ifδis continuous and derivable at X . Also in this year ,Jing Wu has proved on condition that let H be a Hilbert space and B ( H ) stands for the algebra of all bounded linear operators on H , then 0 is a generalized Jordan all-derivable point of B ( H ) if H is infinite-dimensional; for any Hilbert space H , he has also showed that I is a Jordan all-derivable point of B ( H ). Enlightened and guided by these results, we extend the results to the algebra of all 2×2 matrices.In this paper , we have two section , the first one we fix some notation, symbols and fundamental theorems that will be used in this rest of this paper, the second one is the body of the whole paper, we discuss some results of all-derivable points in B ( H ) space, and explore the all- derivable points of all 2×2operator matrices algebra. Throughout this paper ,the nonzero Hilbert space H under consideration are complex and separable, B ( H ) stands for the set of all bounded linear operators from H to H , A be a operator subalgebra of B ( H ), andφbe a linear mappings on A . We sayφ: A→A is a derivation ifφ( ST ) =φ( S ) T + Sφ( T) for ?S ,T∈A. An element P∈A , we say thatφis an derivable mapping at P ifφ( ST ) =φ( S ) T + Sφ( T) for ?S ,T∈A with ST = P. An element P∈A is called an all-derivable point in A if every derivable mapping at P is a derivation. An element P∈A is called an all-derivable point in A for the norm-topology (strong operator topology, etc.) if every norm-topology (strong operator topology, etc.) continuous derivable mapping at P is a derivation. It is the purpose of this paper to prove the following statements, let M stands for the closed subspace of H , and every 2×2 operator matrice is represented relative to the orthogonal decomposition H = M⊕M⊥, with dim M = dim M⊥. Let A be the algebra of all 2×2 operator matrices. i.e. We shall show that if G satisfying one of the following conditions that or ( G is any invertible operator ) are all-derivable points of B ( H ).