Surjective Linear Map Research Articles

Crossed products of 4-algebras. Applications

A 4-algebra is a commutative algebra A over a field k such that (a2)2=0, for all a∈A. We have proved recently [22] that 4-algebras play a prominent role in the classification of finite dimensional Bernstein algebras. Let A be a 4-algebra, E a vector space and π:E→A a surjective linear map with V=Ker(π). All 4-algebra structures on E such that π:E→A is an algebra map are described and classified by a global cohomological object GH2(A,V). Any such 4-algebra is isomorphic to a crossed product V#A and GH2(A,V) is a coproduct, over all 4-algebras structures ⋅V on V, of all non-abelian cohomologies Hnab2(A,(V,⋅V)), which are the classifying objects for all extensions of A by V. Several applications and examples are provided: in particular, GH2(A,k) and GH2(k,V) are explicitly computed and the Galois group Gal(V#A/V) of the extension V↪V#A is described.

Maps preserving two-sided zero products on Banach algebras

Let A and B be Banach algebras with bounded approximate identities and let Φ:A→B be a surjective continuous linear map which preserves two-sided zero products (i.e., Φ(a)Φ(b)=Φ(b)Φ(a)=0 whenever ab=ba=0). We show that Φ is a weighted Jordan homomorphism provided that A is zero product determined and weakly amenable. These conditions are in particular fulfilled when A is the group algebra L1(G) with G any locally compact group. We also study a more general type of continuous linear maps Φ:A→B that satisfy Φ(a)Φ(b)+Φ(b)Φ(a)=0 whenever ab=ba=0. We show in particular that if Φ is surjective and A is a C⁎-algebra, then Φ is a weighted Jordan homomorphism.

Operators preserving mutual strong Birkhoff–James orthogonality on [formula omitted

We classify surjective linear maps on B(H) that preserve mutual strong Birkhoff–James orthogonality.

Maps preserving matrices of extremal scrambling index

Abstract In this paper we characterize surjective linear maps on matrices over antinegative semirings that preserve the set of matrices with maximal or minimal positive values of the scrambling index.

A characterization of an isomorphism

Let [Formula: see text] and [Formula: see text] be some standard operator algebras on complex Banach spaces [Formula: see text] and [Formula: see text], respectively, and [Formula: see text] be a polynomial with no repeated roots and [Formula: see text], such that [Formula: see text]. We characterize the forms of surjective linear maps [Formula: see text] which preserve the nonzero products of operators that annihilated by [Formula: see text].

On maps preserving operators of local spectral radius zero

Let L(X) be the algebra of all bounded linear operators on a complex Banach space X. We describe surjective linear maps ϕ on L(X) that satisfyrϕ(T)(x)=0⟹rT(x)=0 for every x∈X and T∈L(X). We also describe surjective linear maps ϕ on L(X) that satisfyrT(x)=0⟹rϕ(T)(x)=0 for every x∈X and T∈L(X). Furthermore, we characterize maps ϕ (not necessarily linear nor surjective) on L(X) which satisfyrϕ(T)−ϕ(S)(x)=0 if and only ifrT−S(x)=0 for every x∈X and T,S∈L(X).

Linear Preservers of Quadratic Operators

Let \({{\mathcal B}(H)}\) be the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space \({H}\). In this paper, we get a complete classification of surjective linear maps on \({{\mathcal B}(H)}\) that preserve quadratic operators in both directions. An analogue result in the setting of finite-dimensional Banach spaces is given.

Common zeros preserving maps on vector-valued function spaces and Banach modules

Let X, Y be Hausdorff topological spaces, and let E and F be Hausdorff topological vector spaces. For certain subspaces A (X,E) and A(Y, F) of C(X,E) and C(Y, F) respectively (including the spaces of Lipschitz functions), we characterize surjections S, T : A (X;E) → A(Y, F), not assumed to be linear, which jointly preserve common zeros in the sense that Z (f – f’) ∩ Z (f – f’) ∩ Z (g – g’) ≠ 0 if and only if Z (Sf – Sf’) ∩ Z (Tg – Tg´) ≠ 0 for all f, f’, g, g’ ∈ A (X, E). Here Z (·)denotes the zero set of a function. Using the notion of point multipliers we extend the notion of zero set for the elements of a Banach module and give a representation for surjective linear maps which jointly preserve common zeros in module case.

Hochschild products and global non-abelian cohomology for algebras. Applications

Let A be a unital associative algebra over a field k, E a vector space and π:E→A a surjective linear map with V=Ker(π). All algebra structures on E such that π:E→A becomes an algebra map are described and classified by an explicitly constructed global cohomological type object GH2(A,V). Any such algebra is isomorphic to a Hochschild product A⋆V, an algebra introduced as a generalization of a classical construction. We prove that GH2(A,V) is the coproduct of all non-abelian cohomologies H2(A,(V,⋅)). The key object GH2(A,k) responsible for the classification of all co-flag algebras is computed. All Hochschild products A⋆k are also classified and the automorphism groups AutAlg(A⋆k) are fully determined as subgroups of a semidirect product A⁎⋉(k⁎×AutAlg(A)) of groups. Several examples are given as well as applications to the theory of supersolvable coalgebras or Poisson algebras. In particular, for a given Poisson algebra P, all Poisson algebras having a Poisson algebra surjection on P with a 1-dimensional kernel are described and classified.

Global Convergence of Splitting Methods for Nonconvex Composite Optimization

We consider the problem of minimizing the sum of a smooth function $h$ with a bounded Hessian, and a nonsmooth function. We assume that the latter function is a composition of a proper closed function $P$ and a surjective linear map $\cal M$, with the proximal mappings of $\tau P$, $\tau > 0$, simple to compute. This problem is nonconvex in general and encompasses many important applications in engineering and machine learning. In this paper, we examined two types of splitting methods for solving this nonconvex optimization problem: alternating direction method of multipliers and proximal gradient algorithm. For the direct adaptation of the alternating direction method of multipliers, we show that, if the penalty parameter is chosen sufficiently large and the sequence generated has a cluster point, then it gives a stationary point of the nonconvex problem. We also establish convergence of the whole sequence under an additional assumption that the functions $h$ and $P$ are semi-algebraic. Furthermore, we give simple sufficient conditions to guarantee boundedness of the sequence generated. These conditions can be satisfied for a wide range of applications including the least squares problem with the $\ell_{1/2}$ regularization. Finally, when $\cal M$ is the identity so that the proximal gradient algorithm can be efficiently applied, we show that any cluster point is stationary under a slightly more flexible constant step-size rule than what is known in the literature for a nonconvex $h$.

Zero products preserving maps from the Fourier algebra of amenable groups

Let $G$ be a locally compact amenable group. The goal of this paper is to investigate the problem of surjective zero products preserving maps from the Fourier algebra of $G$ into a completely contractive Banach algebra. We show that if $B$ is a completely contractive Banach algebra which is faithful and factors weakly, then every surjective completely bounded linear map from $A(G)$ into $B$ which preserves zero products is a weighted homomorphism. Moreover an equivalent condition is given for such a map to be a homomorphism. In particular, this result implies that if $B$ is a commutative C$^*$-algebra or a matrix space and $T:A(G)\rightarrow B$ is a continuous surjective linear map which preserves zero products, then $T$ is a weighted homomorphism and there is an equivalent condition for $T$ to be a homomorphism.

Linear maps between operator algebras preserving certain spectral functions

Let H be an infinite dimensional complex Hilbert space and let be a surjective linear map on B(H) with (I) I 2 K(H), where K(H) denotes the closed ideal of all compact operators on H. If preserves the set of upper semi-Weyl operators and the set of all normal eigenvalues in both directions, then is an automorphism of the algebra B(H). Also the relation between the linear maps preserving the set of upper semi-Weyl operators and the linear maps preserving the set of left invertible operators is considered.

On linear maps compressing or depressing certain subspaces

Let X be a complex Banach space and letL(X) be the Banach algebra of all bounded linear operators on X. We characterize surjective linear maps :L(X)! L(X) compressing or depressing any one of the range, the hyper-range, the analytic core and the kernel.

Orthogonality preserving linear maps on [formula omitted]-algebras with non-zero socles

We study orthogonality preserving surjective linear maps from a unital C∗-algebra with non-zero socle A to a C∗-algebra. In particular, we prove that, when A has essential socle, every biorthogonality preserving surjective linear map T:A→B is automatically bounded and a Jordan ∗-isomorphism multiplied by an invertible element.

Metric versions of Herstein's theorems on Jordan maps

Let A be a Banach algebra and let B be an ultraprime Banach algebra. If Φ: A → B is a surjective continuous linear map which tends to satisfy the Jordan multiplicativity condition, then we show that Φ comes near to satisfy either the multiplicativity or the anti-multiplicativity condition. In fact, we give a quantitative estimate of this phenomenon. Furthermore, we estimate how much a continuous linear map Δ: B → B approaches to satisfy the derivation identity in the case when Δ tends to satisfy the Jordan derivation identity

Minimum and Surjectivity Moduli Preserving in Banach Space

Let m(T) and q(T) be respectively the minimum and the surjectivity moduli of T??(X), where ?(X) denotes the algebra of all bounded linear operators on a complex Banach space X. If there exists a semi-invertible but non-invertible operator in ?(X) then, given a surjective unital linear map ?:??(X)??(X), we prove that m(T)=m(?(T)) for all T??(X), if and only if, q(T)=q(?(T)) for all T??(X), if and only if, there exists a bijective isometry U??(X) such that ?(T)=UTU ?1 for all T??(X).

LINEAR ORTHOGONALITY PRESERVERS OF STANDARD OPERATOR ALGEBRAS

In 2003, Araujo and Jarosz showed that every bijective linear map $\theta: A \to B$ between unital standard operator algebras preserving zero products in two ways is a scalar multiple of an inner automorphism. Later in 2007, Zhao and Hou showed that similar results hold if both $A,B$ are unital standard algebras on Hilbert spaces and $\theta$ preserves range or domain orthogonality. In particular, such maps are automatically bounded. In this paper, we will study linear orthogonality preservers in a unified way. We will show that every surjective linear map between standard operator algebras preserving range/domain orthogonality carries a standard form, and is thus automatically bounded.

LINEAR ORTHOGONALITY PRESERVERS

In 2003, Araujo and Jarosz showed that every bijective linear map θ : A → B between unital standard operator algebras preserving zero products in two ways is a scalar multiple of an inner automorphism. Later in 2007, Zhao and Hou showed that similar results hold if both A,B are unital standard algebras on Hilbert spaces and θ preserves range or domain orthogonality. In particular, such maps are automatically bounded. In this paper, we will study linear orthogonality preservers in a unified way. We will show that every surjective linear map between standard operator algebras preserving range/domain orthogonality carries a standard form, and is thus automatically bounded.

Linear maps preserving the minimum modulus

We characterize surjective linear maps that preserve the minimum modulus between unital semisimple Banach algebras, one of them is a unital C ∗ -algebra having either real rank zero or essential socle. We also describe surjective linear maps on L(H), with H an infinite- dimensional Hilbert space, preserving the essential minimum modulus. Results concerning sur- jectivity and maximum modulus are also obtained.

Linear maps preserving the minimum and surjectivity moduli of operators

Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space H, and denote by m(T) and q(T) respectively the minimum modulus and the surjectivity mod- ulus for every T ∈ B(H). In this paper, we prove that if φ is a surjective unital linear map on B(H) ,t hen m(φ(T)) = m(T) for every T ∈ B(H) if and only if q(φ(T)) = q(T) for every T ∈ B(H) if and only if there exists an unitary operator U ∈ B(H) such that φ(T )= UTU ∗ for all T ∈ B(H).