Linear contraction mappings
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).
Local behaviour of operators
Local ergodic theorems.
Local spectral theory of shifts with operator-valued weights. (Sur la théorie spectrale locale des shifts à poids opérateurs.)
Local spectrum and Kaplansky's theorem on algebraic operators
Using elementary arguments we improve former results of P. Vrbová concerning local spectrum. As a consequence, we obtain a new proof of Kaplansky’s theorem on algebraic operators on a Banach space.
Local spectrum and local spectral radius of an operator at a fixed vector
Let be a complex Banach space and e ∈ a nonzero vector. Then the set of all operators T ∈ ℒ() with , respectively , 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.