The Properties Of Some Mappings On Operator Algebra

Characterizing linear or additive mapping's preserving problem between operator algebras is one of the most active research topics in the mathematical study during the past century. The most valuable research result is that some authors have showed that a linear or additive mapping can be characterized by an algebraic homomorphism in certain conditions. In the last several dozens years, many authors paid their attention to the study of stability problem, derivability and multiplicative property problem. Many outstanding results have been obtained in these related fields. With the development of the research, more and more results have been gotten. Hou and Qi prove that every derivable mapping at unit operator on J-subspace lattice algebras is a derivation. Zhu Jun and Xiong Changping have proved that every strongly operator topology continuous derivable mapping at an invertible operator on nest algebras is a derivation. Later some authors also proved that a linear mapping which is derivable at some operator is a derivation, even is an inner derivation on the certain conditions. Since S.M.Ulam gave the question concerning the stability of homomorphism in 1940, many scholars had researched about the question. Following the solved questions being increasing, the extensive Hyers-Ulam-Rassias stability problem had been proposed.Let H be a Hilbert space over the field of the complex numbers. B(H) stands for the set of all bounded linear operators from H into H.X is a subalgebra of B(H). We say that a linear mappingφfrom X into itself is multiplicative at G∈X ifφ(AB)=φ(A)φ(B) for any A,B∈X with AB=G; We say that a linear mappingφfrom X into itself is Jordan multiplicative at G∈X ifφ(AB+BA)=φ(A)φ(B)+φ(B)φ(A)φ(AB)=φ(A)φ(B) for any A,B∈X with AB+BA=G. Recently Zhu Jun and Xiong Changping have proved that every irreversible G is a multiplicative point on the matrix algebras M_n.This paper is presented as four chapters. Chapter 1 we mainly introduce some symbols, some definitions and the optional literature's core content which are relevant to this paper. Chapter 2 can be presented two little parts. The first part we show that the Jordan homomorphism's Ulam stability of Cauchy functional equation and Jensen functional equation on the restricted domains of Banach space. The main conclusions are theorem 2.1 and theorem 2.2. Theorem 2.1: Let d>0 andθ>0 be fixed. If a mapping f:X→Y satisfies inequality‖f(x₁+x₂)-f(x₁)-f(x₂)‖≤θfor all x₁,x₂∈Xwith‖x₁‖+‖x₂‖≥d and also satisfies‖f(x₁x₂+x₂x₁)-f(x₁)f)x₂)-f(x₂)f(x₁)‖≤θfor all x₁ ,x₂∈X. If f(sx) is continuous in s∈R for each fixed x∈X, then there exists a unique homomorphism H:X→Y such that‖f(x)-H(x)‖≤4θ+‖f(0)‖for all x∈X. Theorem 2.2: Let d>0 andθ>0 be fixed. If a mapping f:X→Y satisfies inequality‖2f〔(x₁+x₂)/2〕-f(x₁)-f(x₂)‖≤θfor all x₁ ,x₂∈Xwith 1/2(‖x₁‖+‖x₂‖)≥d and also satisfies inequality‖f(x₁x₂+x₂x₁)-f(x₁)f)x₂)-f(x₂)f(x₁)‖≤θfor all x₁ ,x₂∈X. If f(sx) is continuous in s∈R for each fixed x∈X, then there exists a unique homomorphism H:X→Y such that‖f(x)-H(x)‖≤5θ+‖f(0)‖for all x∈X. The second part we prove that the stability of homomorphism via generalized Jensen equation. The main conclusion is theorem 2.3: Let r,s,t be positive numbers and f:X→Y be a mapping for which there exists a functionφ:X×X→[0,+∞) not only satisfying (?) but also satisfying‖f(xy+yx)-f(x)f(y)-f(y)f(x)‖≤φ(x,y) for all x ,y∈X and‖rf〔(sx+ty)/r〕-sf(x)-tf(y)‖≤φ(x,y) for all x ,y∈X with r/(s+t)>1. Then there exists a unique homomorphism H:X→Ygiven by (?) satisfying (?) for all x∈X. Chapter 3 can also be divided into two little parts. In the first part we letΑbe a subalgebra and we show that every linear mappingφfromΑinto itself which satisfiesφ(AB+BA)=φ(A)φ(B)+φ(B)φ(A) at the irreversible point G with AB=G for any A,B∈Αis a multiplicative mapping. In the second part we letΑbe a subalgebra and let the fixed point G be similar to ( )E_nI_n-1 . In this part we show that every linear mappingφfromΑinto itself which satisfiesφ(AB+BA)=φ(A)φ(B)+φ(B)φ(A) at the fixed irreversible point G with AB+BA = G for any A,B∈Αis a Jordan multiplicative mapping. At this time we call the fixed point G is a Jordan all-multiplicative point. Chapter 4 mainly summarizes the full text.