Let H be a complex, separable, infinite dimensional Hilbert space, T ∈ (H). (u+К)(T)denotes the (u+К)-orbit of T, i.e., (u +К)(T) = {R-1TR: R is invertible and of the form unitary plus compact}. Let Ω be an analytic and simply connected Cauchy domain in C and n ∈ N. A(Ω, n) denotes the class of operators, each of which satisfies (i) T is essentially normal; (ii) σ(T) = -Ω, pF(T) ∩σ(T)=Ω; (iii) ind (λ-T) = -n, nul(λ-T)＝0(λ∈Ω). It is proved that given T1, T2∈A(Ω, n) and e＞0, there exists a compact operator K with‖K‖＜e such that T1+K∈(u+К)(T2). This result generalizes a result of P. S. Guinand and L. Marcoux[6,15]. Furthermore, the authors give a character of the norm closure of (u+К)(T),and prove that for each T ∈ A(Ω, n), there exists a compact (SI) perturbation of T whose norm can be arbitrarily small.