In this paper, we describe linear maps between complex Banach algebras that preserve products equal to fixed elements. This generalizes some important special cases where the fixed elements are the ...zero or identity element. First we show that if such map preserves products equal to a finite-rank operator, then it must also preserve the zero product. In several instances, this is enough to show that a product preserving map must be a scalar multiple of an algebra homomorphism. Second, we explore a more general problem concerning the existence of product preserving maps and the relationship between the fixed elements. Lastly, motivated by Kaplansky's problem on invertibility preservers, we show that maps preserving products equal to fixed invertible elements are either homomorphisms or antihomomorphisms multiplied on the left by a fixed element.
Let X be a finite connected poset, F a field and I(X,F) the incidence algebra of X over F. We describe the bijective linear idempotent preservers φ:I(X,F)→I(X,F). Namely, we prove that, whenever ...char(F)≠2, φ is either an automorphism or an anti-automorphism of I(X,F). If char(F)=2 and |F|>2, then φ is a (in general, non-proper) Lie automorphism of I(X,F). Finally, if F=Z2, then φ is the composition of a bijective shift map and a Lie automorphism of I(X,F). Under certain restrictions on the characteristic of F we also obtain descriptions of the bijective linear maps which preserve tripotents and, more generally, k-potents of I(X,F) for k≥3.
We prove that all injective maps on positive complex matrices which preserve order and shrink spectrum are implemented by unitary or antiunitary conjugations. We show by counterexamples that all ...assumptions are indispensable. The result easily generalizes to maps on hermitian matrices.
Let B(X) be the space of all bounded linear operators on complex Banach space X. For A ∈ B(X), we denote by F(A) the subspace of all fixed points of A. In this paper, we study and characterize all ...surjective maps φ on B(X) satisfying F(φ(T)φ(A) + φ(A)φ(T)) = F(T A + AT) for all A, T ∈ B(X).
Let A be a unital C⁎-algebra equipped with its natural order. As usual the effect algebra of A is the interval {a∈A:0≤a≤IA}, where IA denotes the unit of A. In this paper, we give a complete ...description of order isomorphisms between effect algebras of C⁎-algebras of type C(X)⊗B(H), where C(X) stands for the algebra of all continuous complex valued functions on a (non pathological) Hausdorff compact space X and B(H) denotes the algebra of all bounded linear operators on a complex Hilbert space H. Our results generalize some works by L. Molnár and P. Šemrl.
Let $\textbf{M}_{m,n}$ be the set of all $m$-by-$n$ real matrices. A matrix $R$ in $\textbf{M}_{m,n}$ with nonnegative entries is called strictly sub row stochastic if the sum of entries on every ...row of $R$ is less than 1. For $A,B\in\textbf{M}_{m,n}$, we say that $A$ is strictly sub row Hadamard majorized by $B$ (denoted by $A\prec_{SH}B)$ if there exists an $m$-by-$n$ strictly sub row stochastic matrix $R$ such that $A=R\circ B$ where $X \circ Y$ is the Hadamard product (entrywise product) of matrices $X,Y\in\textbf{M}_{m,n}$. In this paper, we introduce the concept of strictly sub row Hadamard majorization as a relation on $\textbf{M}_{m,n}$. Also, we find the structure of all linear operators $T:\textbf{M}_{m,n} \rightarrow \textbf{M}_{m,n}$ which are preservers (resp. strong preservers) of strictly sub row Hadamard majorization.
Let X be an infinite-dimensional complex Banach space, and let B(X) be the algebra of all linear and bounded operators on X. In this paper, we characterize linear bijective maps φ on B(X) having the ...property that the spectral radius (respectively, the spectrum) of φ(A) equals the spectral radius (respectively, the spectrum) of φ(B) for each pair of similar operators A,B∈B(X). As a corollary, we obtain a characterization of linear bijective maps on B(X) preserving the equality of the spectrum.
Abstract Let X $\mathcal {X}$ be an infinite-dimensional complex Banach space, B ( X ) $\mathcal {B}(\mathcal {X})$ the algebra of all bounded linear operators on X $\mathcal {X}$ . Denote the ...spectral domain by σ γ ( T ) = { λ ∈ σ a ( T ) : T that is semi-Fredholm and a s c ( T − λ I ) < ∞ } $\sigma _{\gamma}(T)=\{\lambda \in \sigma _{a}(T): T { \text{ that is semi-Fredholm and }} asc(T-\lambda I)<\infty \}$ . In this paper, we characterize the structure of additive surjective maps φ : B ( X ) → B ( X ) $\varphi : \mathcal {B}(\mathcal {X})\rightarrow \mathcal {B}(\mathcal {X})$ with σ γ ( φ ( T ) ) = σ γ ( T ) $\sigma _{\gamma}(\varphi (T))=\sigma _{\gamma}(T)$ for all T ∈ B ( X ) $T\in \mathcal{B(X)}$ .
Let X X be a complex Banach space, and denote by B ( X ) \mathcal {B}(X) the algebra of all bounded linear operators on X X . Let C , D ∈ B ( X ) C,D\in \mathcal {B} \left ( X\right ) be fixed ...operators. In this paper, we characterize linear, continuous and bijective maps φ \varphi and ψ \psi on B ( X ) \mathcal {B}\left ( X\right ) for which there exist invertible operators T 0 , W 0 ∈ B ( X ) T_0, W_0 \in \mathcal { B}(X) such that φ ( T 0 ) , ψ ( W 0 ) ∈ B ( X ) \varphi (T_0), \psi (W_0) \in \mathcal {B}(X) are both invertible, having the property that φ ( A ) ψ ( B ) = D \varphi \left ( A\right ) \psi \left ( B\right ) =D in B ( X ) \mathcal {B}(X) whenever A B = C AB=C in B ( X ) \mathcal {B}(X) . As a corollary, we deduce the form of linear, bijective and continuous maps φ \varphi on B ( X ) \mathcal {B}(X) having the property that φ ( A ) φ ( B ) = D \varphi \left ( A\right ) \varphi \left ( B\right ) =D in B ( X ) \mathcal {B}(X) whenever A B = C AB=C .
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