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Linear maps Lie derivable at zero on 𝒥-subspace lattice algebras

Xiaofei Qi, Jinchuan Hou (2010)

Studia Mathematica

A linear map L on an algebra is said to be Lie derivable at zero if L([A,B]) = [L(A),B] + [A,L(B)] whenever [A,B] = 0. It is shown that, for a 𝒥-subspace lattice ℒ on a Banach space X satisfying dim K ≠ 2 whenever K ∈ 𝒥(ℒ), every linear map on ℱ(ℒ) (the subalgebra of all finite rank operators in the JSL algebra Alg ℒ) Lie derivable at zero is of the standard form A ↦ δ (A) + ϕ(A), where δ is a generalized derivation and ϕ is a center-valued linear map. A characterization of linear maps Lie derivable...

Linear maps on Mₙ(ℂ) preserving the local spectral radius

Abdellatif Bourhim, Vivien G. Miller (2008)

Studia Mathematica

Let x₀ be a nonzero vector in ℂⁿ. We show that a linear map Φ: Mₙ(ℂ) → Mₙ(ℂ) preserves the local spectral radius at x₀ if and only if there is α ∈ ℂ of modulus one and an invertible matrix A ∈ Mₙ(ℂ) such that Ax₀ = x₀ and Φ ( T ) = α A T A - 1 for all T ∈ Mₙ(ℂ).

Linear maps preserving elements annihilated by the polynomial X Y - Y X †

Jianlian Cui, Jinchuan Hou (2006)

Studia Mathematica

Let H and K be complex complete indefinite inner product spaces, and ℬ(H,K) (ℬ(H) if K = H) the set of all bounded linear operators from H into K. For every T ∈ ℬ(H,K), denote by T † the indefinite conjugate of T. Suppose that Φ: ℬ(H) → ℬ(K) is a bijective linear map. We prove that Φ satisfies Φ ( A ) Φ ( B ) = Φ ( B ) Φ ( A ) † for all A, B ∈ ℬ(H) with A B = B A † if and only if there exist a nonzero real number c and a generalized indefinite unitary operator U ∈ ℬ(H,K) such that Φ ( A ) = c U A U † for all A ∈ ℬ(H).

Linear maps preserving quasi-commutativity

Heydar Radjavi, Peter Šemrl (2008)

Studia Mathematica

Let X and Y be Banach spaces and ℬ(X) and ℬ(Y) the algebras of all bounded linear operators on X and Y, respectively. We say that A,B ∈ ℬ(X) quasi-commute if there exists a nonzero scalar ω such that AB = ωBA. We characterize bijective linear maps ϕ : ℬ(X) → ℬ(Y) preserving quasi-commutativity. In fact, such a characterization can be proved for much more general algebras. In the finite-dimensional case the same result can be obtained without the bijectivity assumption.

Linear maps preserving the generalized spectrum.

Mostafa Mbekhta (2007)

Extracta Mathematicae

Let H be an infinite-dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. For an operator T in B(H), let σg(T) denote the generalized spectrum of T. In this paper, we prove that if φ: B(H) → B(H) is a surjective linear map, then φ preserves the generalized spectrum (i.e. σg(φ(T)) = σg(T) for every T ∈ B(H)) if and only if there is A ∈ B(H) invertible such that either φ(T) = ATA-1 for every T ∈ B(H), or φ(T) = ATtrA-1 for every T ∈ B(H). Also, we...

Linear operators on non-locally convex Orlicz spaces

Marian Nowak, Agnieszka Oelke (2008)

Banach Center Publications

We study linear operators from a non-locally convex Orlicz space L Φ to a Banach space ( X , | | · | | X ) . Recall that a linear operator T : L Φ → X is said to be σ-smooth whenever u ₙ ⟶ ( o ) 0 in L Φ implies | | T ( u ₙ ) | | X → 0 . It is shown that every σ-smooth operator T : L Φ → X factors through the inclusion map j : L Φ → L Φ ̅ , where Φ̅ denotes the convex minorant of Φ. We obtain the Bochner integral representation of σ-smooth operators T : L Φ → X . This extends some earlier results of J. J. Uhl concerning the Bochner integral representation of linear operators defined on a locally convex...

Linearization and compactness

Jesús Ángel Jaramillo, Ángeles Prieto, Ignacio Zalduendo (2009)

Studia Mathematica

This paper is devoted to several questions concerning linearizations of function spaces. We first consider the relation between linearizations of a given space when it is viewed as a function space over different domains. Then we study the problem of characterizing when a Banach function space admits a Banach linearization in a natural way. Finally, we consider the relevance of compactness properties in linearizations, more precisely, the relation between different compactness properties of a mapping,...

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