Linear mappings preserving similarity on B(H)
Let H be an infinite-dimensional complex Hilbert space. We give a characterization of surjective linear mappings on B(H) that preserve similarity in both directions.
Linear maps Lie derivable at zero on 𝒥-subspace lattice algebras
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
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 for all T ∈ Mₙ(ℂ).
Linear maps preserving elements annihilated by the polynomial
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 the indefinite conjugate of T. Suppose that Φ: ℬ(H) → ℬ(K) is a bijective linear map. We prove that Φ satisfies for all A, B ∈ ℬ(H) with if and only if there exist a nonzero real number c and a generalized indefinite unitary operator U ∈ ℬ(H,K) such that for all A ∈ ℬ(H).
Linear maps preserving quasi-commutativity
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.
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 modulus of a multidimensional semigroup of -contractions and a differentiation theorem
Linear neutral partial differential equations: A semigroup approach.
Linear ODEs and -modules, solving and decomposing equations using symmetry methods.
Linear operators
Linear Operators and Vector Measures. II.
Linear operators as measure preserving transformations
Linear operators in the space of regulated functions
Representation of bounded and compact linear operators in the Banach space of regulated functions is given in terms of Perron-Stieltjes integral.
Linear operators on
Linear operators on for dispersed
Linear operators on and Lorentz spaces: The Krasnosel’skii-Zabrieko characteristic sets
Linear operators on non-locally convex Orlicz spaces
We study linear operators from a non-locally convex Orlicz space to a Banach space . Recall that a linear operator is said to be σ-smooth whenever in implies . It is shown that every σ-smooth operator factors through the inclusion map , where Φ̅ denotes the convex minorant of Φ. We obtain the Bochner integral representation of σ-smooth operators . This extends some earlier results of J. J. Uhl concerning the Bochner integral representation of linear operators defined on a locally convex...
Linear positive operators and their applications to differential equations
Linear preservers on ℬ(X)
Linear stability theory of solitary waves arising from Hamiltonian systems with symmetry