Maps Preserving Peripheral Spectrum Of Generalized Product Of Operators On Standard Operator Algebras

General preserver problems is to study the maps leaving some properties of elements in algebras invariant. Often, the characterizations of such preservers imply that they are algebric isomorphisms or algebric anti-isomorphisms, 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 purpose of studying general preserver problems is to seek the rigid invariant of isomorphisms, provide information of the whole structure of operator algebras and classification of operator algebras from a new angle.In this paper, we mainly study characterization of maps between standard opera-tor algebras on complex Banach spaces preserving peripheral spectrum of the generalized products of operators, generalized Jordan product of operators and Lie product of opera-tors.Let A1 and A2 be standard operator algebras on complex Banach spaces X1 and X2, respectively. Denote by σ(T), r(T) and σπ(T) = {λ∈σ(T) | |λ| = r(T)} the spectrum, the spectral radius and the peripheral spectrum of operator T. Let m, k be positive integers with m ≥k≥ 2, and (i1,...,im) be a fixed sequence such that{i1,...,im} = {1,...,k} and at least one of the terms in (i1,..., im) appears exactly once. For a given sequence, and operators T1,..., Tk∈A1, the operators T1oT2...oTk = Ti1Ti2···Tim and T1 * T2 * ? ? ? *Tk = Ti1Ti2 ···Tim + Tim ··· Ti2Ti1 are respectively called generalized product and generalized Jordan product of T1,..., Tk. If X1 is a Hilbert space, skew generalized product and the skew generalized Jordan product for operators T1,..., Tk are defined respectively by T1(?)T2(?)···(?)Tk = Ti1Ti2 ...Tip*…Tim and T1·T2·??? .Tfc=Ti1…T*p…Tim+Tim…T*p…Til.The following are the main results obtained in this thesis:1.Assume that Φ :A1â†'A2 is a map the range of which contains all operators of rank at most two.Then Φ preserves the peripheral spectrum of generalized products, i.e., Φ satisfies that σπ(Φ(A1)o…o Φ(Ak)) = σπ(A1 o…o Ak) for all A1, A2,..., Ak∈A1 if and only if Φ is an isomorphism or an anti-isomorphism multiplied by an mth root of unity. In the last case, the spaces X1 and X2 must be reflexive, A1 o…o Ak a general quasi-semi Jordan product or k = 2. Particularly, if the generalized product is not semi Jordan and k≥3, then Φ preserves the peripheral spectrum of the generalized product if and only if Φ is an isomorphism multiplied by an mth root of 1.2.Let A1 and A2 be standard operator algebras on complex Hilbert spaces H1 and H2, respectively. Assume that Φ :A1â†'A2 is a map the range of which contains all operators of rank at most two. Then Φ preserves peripheral spectrum of the skew generalized product of operators if and only if Φ is *-isomorphisms or * anti-isomorphisms multiplied by a scalar c∈{-1,1}. In the last case, the skew generalized product A1…Ak is quasi-semi Jordan. Moreover, c=1 whenever m is odd.3.Let A1 and A2 be standard real Jordan algebras of self-adjoint operators on complex Hilbert spaces H1 and H2, respectively. Assume that Φ : A1â†'A2 is a map the range of which contains all self-adjoint operators of rank at most two. Then Φ preserves peripheral spectrum of generalized products of self-adjoint operators if and only if Φ are of the form Aâ†'cU Au* or Aâ†'cU AtU*, where U ∈B(H1,H2) is a unitary operator, c∈{1,-1}. Moreover, the range of Φ contains all rank one projections and c = 1 whenever m is odd.4. Let X1 and X2 be complex Banach spaces with dimension at least three, A1 and A2 be standard operator algebras on X1 and X2, respectively. Assume that Φ : A1 â†' A2 is a map the range of which contains all operators of rank at most three. Then Φ preserves the peripheral spectrum of generalized Jordan products, i.e., Φ satisfies that σπ(Φ(A1) * ··· * Φ(Ak)) = σπ(A1 * …* Ak) for all A1,..., Ak ∈A1 if and only if Φ is a Jordan isomorphism multiplied by an mth root of unity. In the last case, the spaces X1 and X2 must be reflexive.5.Let A1 and A2 be standard real Jordan algebras of self-adjoint operators on complex Hilbert spaces H1 and H2, respectively. Assume that Φ : A1â†'A2 is a map the range of which contains all rank one projections and trace zero rank two self-adjoint operators. Then Φ preserves peripheral spectrum of generalized Jordan products of self-adjoint operators if and only if Φ are of the form Aâ†'cUAU* or Aâ†'cU AtU*, where U∈B(H1,H2) is a unitary operator, c∈{1,-1}. Moreover, c=1 whenever m is odd.6. Let X1 and X2 be complex Banach spaces with dimension at least three, A1 and A2 be standard operator algebras on X\ and X2, respectively. Assume that $ : A1â†'A2 is a additive map the range of which contains all operators of rank at most two and the identity operator. Then Φ preserves the peripheral spectrum of Lie products, i.e.,Φ satisfies that σπ{AB - BA) = σπ(Φ(A)Φ(B)-Φ(B)Φ(A)) for all A, B∈A1 if and only if there exist an invertible T ∈B(X1,X2) satisfying TA1T-1(?) A2 and an additive functional h : A1â†' C such that Φ(A) = λTAT-1 + h{A)I for all A e A1, where A ∈{-1,1}.