Local spectral properties for operators satisfying A nBA n = U (t)A j U (t) ∗

Abstract

In this paper, we investigate the local spectral properties of bounded linear operators A and B acting on a Hilbert space H , which satisfy the generalized operator equation A nBA n = U (t)A j U (t) ∗ , t ∈ I ⊆ R, where U (t) is a strongly continuous family of unitary operators on L . The strong continuity condition means that for every vector x ∈ L , ∥U (t)x− U (s)x∥ → 0 as t → s. We analyze how this time-dependent unitary intertwining affects the transmission of local spectral properties between A and B. In particular, we establish conditions under which certain local spectral properties of A are inherited by B, and vice versa. Several concrete examples involving shift operators, diagonal matrices, and multiplication operators on L 2 spaces are provided in Section 3 to demonstrate the applicability of the developed results and to highlight phenomena arising from the unitary perturbation parameter t.