Skew-product and peripheral local spectrum preservers

Abstract

Let H and K be two infinite-dimentional complex Hilbert spaces, and fix two nonzero vectors h₀ ∈ H and k₀ ∈ K. Let L (H) (resp. L (K)) denote the algebra of all bounded linear operators on H (resp. on K), and let γ_T(x) be the peripheral local spectrum of an operator T at x, and let F_n(K) be the ideal of all operators in L (K) of rank at most n. We show that if the maps φ₁ : L(H) → L(K) and φ₂ : L(H) → L(K) satisfy
                      γ_TS*(h₀) = γ_{φ1(T )φ2(S)*} (k₀)
for all T, S ∈ L (H), and their range containing F₂(K), then there exist bijective linear operators U : H → K and V : K → H such that φ₁(T) = UTV and φ₂(T)*= V⁻¹T*U⁻¹ for all T ∈ L(H). Furthermore,
some interesting results are obtained in this direction.