Maps preserving peripheral spectrum of Jordan semi-triple products of operators

Let A and B be (not necessarily unital or closed) standard operator algebras on complex Banach spaces X and Y, respectively. For a bounded linear operator A on X, the peripheral spectrum σ_π(A) of A is the set σ_π(A)={z∈σ(A):|z|=max_ω∈σ(A)|ω|}, where σ(A) denotes the spectrum of A. Assume that Φ:A→B is a map the range of which contains all operators of rank at most two. It is shown that the map Φ satisfies the condition that σ_π(BAB)=σ_π(Φ(B)Φ(A)Φ(B)) for all A,B∈A if and only if there exists a scalar λ∈C with λ³=1 and either there exists an invertible operator T∈B(X,Y) such that Φ(A)=λTAT^-1 for every A∈A; or there exists an invertible operator T∈B(X^∗,Y) such that Φ(A)=λTA^∗T^-1 for every A∈A. If X=H and Y=K are complex Hilbert spaces, the maps preserving the peripheral spectrum of the Jordan skew semi-triple product BA^∗B are also characterized. Such maps are of the form A?UAU^∗ or A?UA^tU^∗, where U∈B(H,K) is a unitary operator, A^t denotes the transpose of A in an arbitrary but fixed orthonormal basis of H.     

Maps completely preserving spectral functions

Let X,Y be infinite dimensional complex Banach spaces and A,B be standard operator algebras on X and Y, respectively. In this paper, we show that surjective maps completely preserving certain spectral function Δ(·) from A to B are isomorphisms, where Δ(·) stands for any one of 13 spectral functions σ(·), σ_l(·), σ_r(·), σ_l(·)∩σ_r(·), ∂σ(·), ησ(·), σ_p(·), σ_c(·), σ_p(·)∩σ_c(·), σ_p(·)∪σ_c(·), σ_ap(·), σ_s(·), and σ_ap(·)∩σ_s(·).     

Maps preserving the nilpotency of products of operators

Let B(X) be the algebra of all bounded linear operators on the Banach space X, and let N(X) be the set of nilpotent operators in B(X). Suppose ?:B(X)→B(X) is a surjective map such that A,B∈B(X) satisfy AB∈N(X) if and only if ?(A)?(B)∈N(X). If X is infinite dimensional, then there exists a map f:B(X)→C?{0} such that one of the following holds:

(a)

There is a bijective bounded linear or conjugate-linear operator S:X→X such that ? has the form A?S[f(A)A]S^-1.

(b)

The space X is reflexive, and there exists a bijective bounded linear or conjugate-linear operator S : X′ → X such that ? has the form A ? S[f(A)A′]S⁻¹.

If X has dimension n with 3 ? n < ∞, and B(X) is identified with the algebra M_n of n × n complex matrices, then there exist a map f:M_n→C?{0}, a field automorphism ξ:C→C, and an invertible S ∈ M_n such that ? has one of the following forms:

     

Maps preserving the idempotency of products of operators

Let B(X) be the algebra of all bounded linear operators on a complex Banach space X. We give the concrete form of every unital surjective map φ on B(X) such that AB is a non-zero idempotent if and only if φ(A)φ(B) is for all A,B∈B(X) when the dimension of X is at least 3.     

Maps preserving operator pairs whose products are projections

Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space H with dimH?2. It is proved that a surjective map φ on B(H) preserves operator pairs whose products are nonzero projections in both directions if and only if there is a unitary or an anti-unitary operator U on H such that φ(A)=λU^∗AU for all A in B(H) for some constants λ with λ²=1. Related results for surjective maps preserving operator pairs whose triple Jordan products are nonzero projections in both directions are also obtained. These show that the operator pairs whose products or triple Jordan products are nonzero projections are isometric invariants of B(H).     

Maps on states preserving the relative entropy II

Let H be a finite-dimensional complex Hilbert space. The aim of this paper is to prove that every transformation on the space of all density operators on H which preserves the relative entropy is implemented by either a unitary or an antiunitary operator on H.     

Maps preserving peripheral spectrum of generalized Jordan products of operators

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. For k ≥ 2, let(i1, i2,..., im) be a finite sequence such that {i1, i2,..., im} = {1, 2,..., k} and assume that at least one of the terms in(i1,..., im) appears exactly once. Define the generalized Jordan product T1 o T2 o ··· o Tk= Ti1Ti2··· Tim+ Tim··· Ti2Ti1 on elements in Ai. This includes the usual Jordan product A1A2 + A2A1, and the Jordan triple A1 A2 A3 + A3 A2 A1. Let Φ : A1 → A2 be a map with range containing all operators of rank at most three. It is shown that Φ satisfies that σπ(Φ(A1) o ··· o Φ(Ak)) = σπ(A1 o ··· o Ak) for all A1,..., Ak,where σπ(A) stands for the peripheral spectrum of A, if and only if Φ is a Jordan isomorphism multiplied by an m-th root of unity.     

Linear maps preserving the essential spectral radius

Let L(H) be the algebra of all bounded linear operators on an infinite dimensional complex Hilbert H. We characterize linear maps from L(H) onto itself that preserve the essential spectral radius.     

On Maps Preserving Zero Jordan Products

Let R be a ring, A = M _n (R) and θ: A → A a surjective additive map preserving zero Jordan products, i.e. if x,y ∈ A are such that xy + yx = 0, then θ(x)θ(y) + θ(y)θ(x) = 0. In this paper, we show that if R contains and n ≥ 4, then θ = λϕ, where λ = θ(1) is a central element of A and ϕ: A → A is a Jordan homomorphism. The third author is Corresponding author.     

Maps leaving functional values of operator products invariant

Let A be a standard operator algebra on a complex Hilbert space H of dimension greater than 2. By invariants of certain functional values of operator products, we characterize some surjective maps on A. Furthermore, several kinds of general preserver problems on standard operator algebras are solved when we take respectively the functional as, for example, k-numerical radius (k?1), operator norm, Ky Fan k-norm, Schatten p-norm (1?p<∞), and so on.     

Additive local spectrum compressors

Let L(X) be the algebra of all bounded linear operators on an infinite dimensional complex Banach space X. We characterize additive continuous maps from L(X) onto itself which compress the local spectrum and the convexified local spectrum at a nonzero fixed vector. Additive continuous maps from L(X) onto itself that preserve the local spectral radius at a nonzero fixed vector are also characterized.     

Surjective linear maps preserving local spectra

Let B(X) be the algebra of all bounded linear operators on an infinite dimensional complex Banach space X. We characterize linear surjective and continuous maps on B(X) preserving different local spectral quantities at a nonzero fixed vector.     

Some generalized theorems onp-hyponormal operators

A bounded linear operatorT is calledp-Hyponormal if (T ^*T)^p(TT ^*)^p, 0<p1. In Aluthge [1], we studied the properties of p-hyponormal operators using the operator . In this work we consider a more general operator , and generalize some properties of p-hyponormal operators obtained in [1].     

The invertibility of the product of unbounded Toeplitz operators

In this note I give necessary and sufficient conditions on outer functionsf andg for the operator to be bounded and invertible on H². I also discuss the relationship of this question to two open questions in operator theory and weighted norm inequalities.     

On the essential approximate point spectrum of operators

If belongs to the essential approximate point spectrum of a Banach space operatorTB(X) and is a sequence of positive numbers with lim ^j a _j =0, then there existsxX such that for every polynomialp. This result is the best possible — if for some constantc>0 thenT has already a non-trivial invariant subspace, which is not true in general.     

Additive maps derivable or Jordan derivable at zero point on nest algebras

Let AlgN be a nest algebra associated with the nest N on a (real or complex) Banach space X. Assume that every N∈N is complemented whenever N_-=N. Let δ:AlgN→AlgN be an additive map. It is shown that the following three conditions are equivalent: (1) δ is derivable at zero point, i.e., δ(AB)=δ(A)B+Aδ(B) whenever AB=0; (2) δ is Jordan derivable at zero point, i.e., δ(AB+BA)=δ(A)B+Aδ(B)+Bδ(A)+δ(B)A whenever AB+BA=0; (3) δ has the form δ(A)=τ(A)+cA for some additive derivation τ and some scalar c. It is also shown that δ is generalized derivable at zero point, i.e., δ(AB)=δ(A)B+Aδ(B)-Aδ(I)B whenever AB=0, if and only if δ is an additive generalized derivation. Finer characterizations of above maps are given for the case dimX=∞.