Research On Additive Mapping Preserving Anti-orthogonality And Preserving Commutative Zero-product

Preserver problems on operator algebras are to study the maps leaving some properties of elements in algebras invariant. Often, the characterizations of such preservers imply that they are algebraic holomorphisms or anti-algebraic holomorphisms, and therefore reveal the connection between the inherent properties of operator algebras and maps on itself. This makes one know and understand operator algebras more deeply. The study of preserver problems not only enrich theory of operator algebras and functional analysis, but also has its practical value in quantum mechanics and system theories. As B(X) is one of the most fundamental operator algebras, the preserver problems on B(X) is the research foundation of the similar problems on general operator algebras. This thesis is exactly to study additive maps preserving anti-orthogonality and additive maps preserving commutative zero-product on B(X). we obtain the following results.In the first place, it is considered that additive maps preserving anti-orthogonality and preserving Jordan orthogonality between the algebras B(H) and B(K) of all bounded linear operators, acting on the real or the complex Hilbert space H and H. Let Φ : B(H) → B(K) be a unital additive surjective map preserving anti-orthogonality in both directions, Φ(FP) (?) FΦ(P) for all rank one idempotent operator, then Φ is a *— anti-isomorphism or a conjugate *— anti-isomorphism. Preserving Jordan orthogonality is characterized under the same assumption as preserving anti-orthogonality, Φ has one of the following forms: a *— isomorphism, a conjugate *— isomorphism, a *— anti-isomorphism, a conjugate *— anti-isomorphism.Secondly, we discuss additive commutative zero-product preserving maps on the algebra B(H) of all bounded linear operators, acting on a real or complex Hilbert space H. If H. is finite dimensional and Φ is a surjective commutative zero-product preserving additive maps on B(H) such that Φ(I) = I and Φ(FP) (?) FΦ(P) for all rank one idempotent operator, then Φ is an automorphism or an anti-automorphism. If H is infinite dimensional and Φ is a surjective commutative zero-product preserving additive maps on B(H), we prove that Φ is a nonzero scalar multiple of a ring isomorphism or a ring anti-isomorphism.Finally, additive commutative zero-product preserving maps on the algebraA, B of standard operator algebras, acting on real or complex Banach spaces X Assume that $ : A —> B is an unital additive surjective map. If $ preserves commutative zero-products and <3>(FP) C F$(P)and $(P) 7^ 0 for every rank one idempotent operator P 6 A. Then it has one of the following forms: an algebraic isomorphism, a conjugate algebraic isomorphism, an algebraic anti-isomorphism and a conjugate algebraic anti-isomorphism.