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.

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 $\Phi \left(T\right)=\alpha AT{A}^{-1}$ for all T ∈ Mₙ(ℂ).

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.

Let be a complex Banach space and e ∈ a nonzero vector. Then the set of all operators T ∈ ℒ() with ${\sigma }_T\left(e\right)={\sigma }_{\delta }\left(T\right)$, respectively $r_T\left(e\right)=r\left(T\right)$, is residual. This is an analogy to the well known result for a fixed operator and variable vector. The results are then used to characterize linear mappings preserving the local spectrum (or local spectral radius) at a fixed vector e.

Let $ℒ\left(\mathscr{H}\right)$ be the algebra of all bounded linear operators on a complex Hilbert space $\mathscr{H}$. We characterize locally spectrally bounded linear maps from $ℒ\left(\mathscr{H}\right)$ onto itself. As a consequence, we describe linear maps from $ℒ\left(\mathscr{H}\right)$ onto itself that compress the local spectrum.