On maps preserving the spectrum of the skew Lie product of operators

Abstract

problem of describing maps on operators and matrices that preserve certain functions, subsets and relations has been widely studied in the literature, see [7], [9], [10], [11], [14], [15], [16], [17], [23], [29], [30], [32], [33] and their references therein. One of the classical problems in this area of research is to characterize maps preserving the spectra of the product of operators. Moln´ar in [29] studied maps preserving the spectrum of operator and matrix products. His results have been extended in several directions [8], [1], [2], [12], [13], [19], [21], [22], [24] and [25]. In [2], the problem of characterizing maps between matrix algebras preserving the spectrum of polynomial products of matrices is considered. In particular, the results obtained therein extend and unify several results obtained in [11] and [13]. Let H and K be two complex infinite dimensional Hilbert spaces. Let B(H) (resp. B(K)) denote the algebra of all bounded linear operators on H (resp. on K ). We say that a map ϕ : B(H) →B(K) preserves the skew Lie product of operators if [ϕ(T),ϕ(S)] ∗ = [T,S] ∗ where [T,S] ∗ = TS −ST∗ for any operators S,T ∈B(H). Latter in [1], the form of all maps preserving the spectrum and the local spectrum of skew Lie product of matrices are determined. In this thesis we will examine the form of surjective maps preserving the spectrum of skew Lie product of operators