Maps preserving the spectrum of generalized Jordan product of operators

Let A₁,A₂ be standard operator algebras on complex Banach spaces X₁,X₂, respectively. For k?2, let (i₁,…,i_m) be a sequence with terms chosen from {1,…,k}, and define the generalized Jordan product

     

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 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).     

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.     

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 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 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 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.     

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.     

Double Operator Integrals in a Hilbert Space

Double operator integrals are a convenient tool in many problems arising in the theory of self-adjoint operators, especially in the perturbation theory. They allow to give a precise meaning to operations with functions of two ordered operator-valued non-commuting arguments. In a different language, the theory of double operator integrals turns into the problem of scalarvalued multipliers for operator-valued kernels of integral operators.The paper gives a short survey of the main ideas, technical tools and results of the theory. Proofs are given only in the rare occasions, usually they are replaced by references to the original papers. Various applications are discussed.     

The higher-order derivatives of spectral functions

We are interested in higher-order derivatives of functions of the eigenvalues of real symmetric matrices with respect to the matrix argument. We describe a formula for the k-th derivative of such functions in two general cases.The first case concerns the derivatives of the composition of an arbitrary (not necessarily symmetric) k-times differentiable function with the eigenvalues of symmetric matrices at a symmetric matrix with distinct eigenvalues.The second case describes the derivatives of the composition of a k-times differentiable separable symmetric function with the eigenvalues of symmetric matrices at an arbitrary symmetric matrix. We show that the formula significantly simplifies when the separable symmetric function is k-times continuously differentiable.As an application of the developed techniques, we re-derive the formula for the Hessian of a general spectral function at an arbitrary symmetric matrix. The new tools lead to a shorter, cleaner derivation than the original one.To make the exposition as self contained as possible, we have included the necessary background results and definitions. Proofs of the intermediate technical results are collected in the appendices.     

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.     

Similarity preserving linear maps on upper triangular matrix algebras

In this note we consider similarity preserving linear maps on the algebra of all n × n complex upper triangular matrices T_n. We give characterizations of similarity invariant subspaces in T_n and similarity preserving linear injections on T_n. Furthermore, we considered linear injections on T_n preserving similarity in M_n as well.     

Linear Maps Preserving Generalized Invertibility

Let H be an infinite-dimensional complex separable Hilbert space and the algebra of all bounded linear operators on H. Let be a bijective continuous unital linear map preserving generalized invertibility in both directions. Then the ideal of all compact operators is invariant under ϕ and the induced linear map on the Calkin algebra is either an automorphism or an antiautomorphism.     

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=∞.     

Bounded characteristic functions and models for noncontractive sequences of operators

The Sz.-Nagy-FoiaŞ functional model for completely non-unitary contractions is extended to completely non-coisometric sequences of bounded operatorsT = (T₁,...,T _d) (d finite or infinite) on a Hilbert space, with bounded characteristic functions. For this class of sequences, it is shown that the characteristic function θ_T is a complete unitary invariant. We obtain, as the main result, necessary and sufficient conditions for a bounded multi-analytic operator on Fock spaces to coincide with the characteristic function associated with a completely non-coisometric sequence of bounded operators on a Hilbert space. Research supported in part by a COBASE grant from the National Research Council. The first author was partially supported by a grant from Ministerul Educaţiei Şi Cercetarii. The second author was partially supported by a National Science Foundation grant.     

Essential spectra of spectral operators

The various essential spectra of a linear operator have been surveyed byB. Gramsch andD. Lay [4]. In this paper we characterize the essential spectra and the related quantities nullity, defect, ascent and descent of bounded spectral operators. It is shown that a number of these spectra coincide in the case of a spectral or a scalar type operator. Some results known for normal operators in Hilbert space are extended to spectral operators in Banach space.     

Isometric Equivalence of Certain Operators on Banach Spaces

A pair of operators on a Banach space X are isometrically equivalent if they are intertwined by a surjective isometry of X. We investigate the isometric equivalence problem for pairs of operators on specific types of Banach spaces. We study weighted shifts on symmetric sequence spaces, elementary operators acting on an ideal I of Hilbert space operators, and composition operators on the Bloch space. This last case requires an extension of known results about surjective isometries of the Bloch space.