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.
