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Linear extenders and the Axiom of Choice

Marianne Morillon (2017)

Commentationes Mathematicae Universitatis Carolinae

In set theory without the Axiom of Choice ZF, we prove that for every commutative field 𝕂 , the following statement 𝐃 𝕂 : “On every non null 𝕂 -vector space, there exists a non null linear form” implies the existence of a “ 𝕂 -linear extender” on every vector subspace of a 𝕂 -vector space. This solves a question raised in Morillon M., Linear forms and axioms of choice, Comment. Math. Univ. Carolin. 50 (2009), no. 3, 421-431. In the second part of the paper, we generalize our results in the case of spherically...

Linear maps preserving A -unitary operators

Abdellatif Chahbi, Samir Kabbaj, Ahmed Charifi (2016)

Mathematica Bohemica

Let ℋ be a complex Hilbert space, A a positive operator with closed range in ℬ ( ℋ ) and ℬ A ( ℋ ) the sub-algebra of ℬ ( ℋ ) of all A -self-adjoint operators. Assume φ : ℬ A ( ℋ ) onto itself is a linear continuous map. This paper shows that if φ preserves A -unitary operators such that φ ( I ) = P then ψ defined by ψ ( T ) = P φ ( P T ) is a homomorphism or an anti-homomorphism and ψ ( T ♯ ) = ψ ( T ) ♯ for all T ∈ ℬ A ( ℋ ) , where P = A + A and A + is the Moore-Penrose inverse of A . A similar result is also true if φ preserves A -quasi-unitary operators in both directions such that there exists an...

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...

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