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).     

An algebraic invariant for Jordan automorphisms on β(<Emphasis Type="Italic">H</Emphasis>): The set of idempotents

Let H be an infinite dimensional complex Hilbert space. Denote by B(H) the algebra of all bounded linear operators on H, and by I(H) the set of all idempotents in B(H). Suppose that Φ is a surjective map from B(H) onto itself. If for every λ ∈ -1,1,2,3, and A, B ∈ B(H),A-λB ∈I(H) ⇔ Φ(A) -λΦ(B) ∈I(H, then Φ is a Jordan ring automorphism, i.e. there exists a continuous invertible linear or conjugate linear operator T on H such that Φ(A) = TAT ^-1 for all A ∈ B(H), or Φ(A) = TA*T ^-1 for all A ∈ B(H); if, in addition, A-iB ∈I(H)⇔ Φ(A)-iΦ(B) ∈I(H), here i is the imaginary unit, then Φ is either an automorphism or an anti-automorphism.     

On linear maps preserving generalized invertibility and related properties

Let H be an infinite-dimensional complex Hilbert space, B(H) be the algebra of all bounded linear operators on H. We study surjective linear maps on B(H) preserving generalized invertibility. We also investigate surjective linear maps preserving Fredholm (respectively, semi-Fredholm) operators. Our results improve those of Mbekhta, Rodman and Šemrl.     

Additive Maps Preserving Nilpotent Operators or Spectral Radius

Let X be a （real or complex） Banach space with dimension greater than 2 and let B0（X） be the subspace of B（X） spanned by all nilpotent operators on X. We get a complete classification of surjective additive maps Ф on B0（X） which preserve nilpotent operators in both directions. In particular, if X is infinite-dimensional, we prove that Ф has the form either Ф（T） = cATA^-1 or Ф（T） = cAT＇A^-1, where A is an invertible bounded linear or conjugate linear operator, c is a scalar, T＇ denotes the adjoint of T. As an application of these results, we show that every additive surjective map on B（X） preserving spectral radius has a similar form to the above with ｜c｜ = 1.     

保持算子乘积投影的线性映射

设B(H)是复Hilbert空间H上的有界线性算子全体且dim H≥2.本文证明了B(H)上的线性满射φ保持两个算子乘积非零投影性的充分必要条件是存在B(H)中的酉算子U以及复常数λ满足λ~2=1,使得φ(X)=λU~*XU,(?)X∈B(H).同时也得到了线性映射保持两个算子Jordan三乘积非零投影的充分必要条件.     

Nonlinear Preserver Problems on B（H）

Let H be a complex Hilbert space of dimension greater than 2, and B(H) denote the Banach algebra of all bounded linear operators on H. For A, B ∈ B(H), define the binary relation A ≤* B by A*A = A*B and AA* = AB*. Then (B(H), “≤*”) is a partially ordered set and the relation “≤*” is called the star order on B(H). Denote by Bs(H) the set of all self-adjoint operators in B(H). In this paper, we first characterize nonlinear continuous bijective maps on B _s (H) which preserve the star order in both directions. We characterize also additive maps (or linear maps) on B(H) (or nest algebras) which are multiplicative at some invertible operator.     

Mappings preserving the idempotency of products of operators

We obtain a general form of a surjective (not assumed additive) mapping φ, preserving the nonzero idempotency of a certain product, being defined (a) on the algebra of all bounded linear operators B(X), where X is at least three-dimensional real or complex Banach space, (b) on the set of all rank-one idempotents in B(X) and (c) on the set of all idempotents in B(X). In any of the cases it turns out that φ is additive and either multiplicative or antimultiplicative.     

Compact differences of composition operators on Holomorphic Function Spaces in the unit ball

We find a lower bound for the essential norm of the difference of two composition operators acting on H 2(BN ) or As2 (BN ) (s > 1). This result plays an important role in proving a necessary and sufficient condition for the difference of linear fractional composition operators to be compact, which answers a question posed by MacCluer and Weir in 2005.     

Isometries of JB-algebras

A bijective linear mapping between two JB-algebrasA andB is an isometry if and only if it commutes with the Jordan triple products ofA andB. Other algebraic characterizations of isometries between JB-algebras are given. Derivations on a JB-algebraA are those bounded linear operators onA with zero numerical range. For JB-algebras of selfadjoint operators we have: IfH andK are left Hilbert spaces of dimension ≥3 over the same fieldF (the real, complex, or quaternion numbers), then every surjective real linear isometryf fromS(H) ontoS(K) is of the formf(x)=UoxoU ⁻¹ forx inS(H), whereτ is a real-linear automorphism ofF andU is a real linear isometry fromH ontoK withU(λh)=τ(λ)U(h) for λ inF andh inH. Supported by Acción Integrada Hispano-Alemana HA 94 066 B     

Lefschetz fixed point theory on fibrations

For a fibre preserving map ϕ: E → E on a fibration (E, π, B), we construct a grading preserving map T(ϕ, π) between H*(E) and H*(B) that generalizes the Lefschetz number. If T(ϕ, π) is an isomorphism between H ⁰(E) and H ⁰(B), then π restricts to a surjective local diffeomorphism on each connected component of the fixed point set of ϕ under a transversality condition. This yields a characterization for the bundle H → G → G/H to be trivial when π ₁ (G/H) = 0.     

A new characterization of Anderson’s inequality in <Emphasis Type="Italic">C</Emphasis><Subscript>1</Subscript>-classes

Let ℋ be a separable infinite dimensional complex Hilbert space, and let ℒ(H) denote the algebra of all bounded linear operators on ℋ into itself. Let A = (A ₁, A ₂,..., A _n), B = (B ₁, B ₂,..., B _n) be n-tuples of operators in ℒ(H); we define the elementary operators Δ _A,B : ℒ(H) ↦ ℒ(H) by

Isometries of non-self-adjoint operator algebras

We study isometries of certain non-self-adjoint operator algebras by means of the structure of the complete holomorphic vector fields on their unit balls and the associated partial Jordan triple products. We show that isometries of nest sub-algebras of B(H) are of the form T UTW or T UJT^*JW, where U, W are suitable unitary operators and J a fixed involution of H.     

2-Summing operators on C([0, 1], lp) with values in l1

Let Ω be a compact Hausdorff space, X a Banach space, C(Ω, X) the Banach space of continuous X-valued functions on Ω under the uniform norm, U: C(Ω, X) → Y a bounded linear operator and U ^#, U _# two natural operators associated to U. For each 1 ≤ s < ∞, let the conditions (α) U ∈ Π _s (C(Ω, X), Y); (β)U ^# ∈ Π _s (C(Ω), Π _s (X, Y)); (γ) U _# ε Π _s (X, Π _s (C(Ω), Y)). A general result, [10, 13], asserts that (α) implies (β) and (γ). In this paper, in case s = 2, we give necessary and sufficient conditions that natural operators on C([0, 1], l _p ) with values in l ₁ satisfies (α), (β) and (γ), which show that the above implication is the best possible result.     

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 \frac12\frac{1}{2} and n ≥ 4, then θ = λϕ, where λ = θ(1) is a central element of A and ϕ: A → A is a Jordan homomorphism.     

Jordan triple derivations on J-subspace lattice algebras

Let L be a J-subspace lattice on a Banach space X and Alg L the associated J-subspace lattice algebra. Let A be a standard operator subalgebra (i.e., it contains all finite rank operators in AlgL) of AlgL and M■B(X) the Alg L-bimodule. It is shown that every linear Jordan triple derivation from A into M is a derivation, and that every generalized Jordan (triple) derivation from A into M is a generalized derivation.     

Additive Maps Preserving Similarity of Operators on Banach Spaces

Let X be an infinite-dimensional complex Banach space and denote by B（X） the algebra of all bounded linear operators acting on X. It is shown that a surjective additive map Φ from B（X） onto itself preserves similarity in both directions if and only if there exist a scalar c, a bounded invertible linear or conjugate linear operator A and a similarity invariant additive functional ψ on B（X） such that either Φ（T） = cATA^-1 ＋ ψ（T）I for all T, or Φ（T） = cAT＊A^-1 ＋ ψ（T）I for all T. In the case where X has infinite multiplicity, in particular, when X is an infinite-dimensional Hilbert space, the above similarity invariant additive functional ψ is always zero.     

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.     

A generalization of peripherally-multiplicative surjections between standard operator algebras

Let A and B be standard operator algebras on Banach spaces X and Y, respectively. The peripheral spectrum σ_π (T) of T is defined by σ_π (T) = z ∈ σ(T): |z| = max_w∈σ(T) |w|. If surjective (not necessarily linear nor continuous) maps φ, ϕ: A → B satisfy σ_π (φ(S)ϕ(T)) = σ_π (ST) for all S; T ∈ A, then φ and ϕ are either of the form φ(T) = A ₁ TA ₂ ⁻¹ and ϕ(T) = A ₂ TA ₁ ⁻¹ for some bijective bounded linear operators A ₁; A ₂ of X onto Y, or of the form φ(T) = B ₁ T*B ₂ ⁻¹ and ϕ(T) = B ₂ T*B ⁻¹ for some bijective bounded linear operators B ₁;B ₂ of X* onto Y.      

Three classes of smooth Banach submanifolds in B(E,F)

Let E, F be two Banach spaces, and B(E, F), Φ(E, F), SΦ(E, F) and R(E,F) be the bounded linear, Fredholm, semi-Frdholm and finite rank operators from E into F, respectively. In this paper, using the continuity characteristics of generalized inverses of operators under small perturbations, we prove the following result Let ∑ be any one of the following sets {T ∈ Φ(E, F) IndexT =const, and dim N(T) = const.}, {T ∈ SΦ(E, F) either dim N(T) = const. ＜ ∞ or codim R(T) = const.＜ ∞} and {T ∈ R(E, F) RankT =const.＜∞}. Then ∑ is a smooth submanifold of B(E, F) with the tangent space TA∑ = {B ∈ B(E,F) BN(A) (∪) R(A)} for any A ∈ ∑. The result is available for the further application to Thom's famous results on the transversility and the study of the infinite dimensional geometry.