On the right essential spectra of quasisimilar operators

In this article, we prove that if T and S are quasisimilar operators, then each nonempty closed-and-open subset of σ_re(T) intersects σ_N(T) ∩ σ_N(S). As a simple corollary we give an affirmative answer to a question of Fialkow. Let L(X) denote the algebra of all operators acting on the complex, infinite dimensional Banach space X. Recall that T ∈ L(X) and S ∈ L(Y) are quasisimilar if there exist injective operators A : X → Y and B : Y → X with dense range such that AT = SA and TB= BS. For T ∈ L(X), we write nul(T) = dimker(T), nul(T*) = dimker(T*) and ind(T) = nul(T) - nul(T*) if at least one of nul(T) and nul(T*) is finite. Recall that A ∈ L(X) is a Semi-Fredholm operator if ran(A) is closed and at least one of nul(A) and nul(A*) is finite, and that A ∈ L(X) is Fredholm operator if ran(A) is closed and both nul(A) and nul(A*) are finite. For T ∈ L(X), the essential spectrum σ_e(T) = {λ ∈ C : T - λ is not Fredholm}, the right essential spectrum σ_re(T) = {λ ∈ C : either T - λ is not Semi-Fredholm or ind(T - λ) = -∞} and left essential spectrum σ_le(T) ={λ ∈ C : either T - λ is not Semi-Fredholm or ind(T - λ) = ∞}. Let σ_lre(T) = {λ ∈ C : T - λ is not Semi-Fredholm}, ψ(T) = {λ ∈ C : ran(T - λ) is closed and ind(T - λ) = n} (n = 0, ±1, ..., ±∞) H_n(T) = {A ∈ C: ind(T - λ) = n} (n = 0, ±1, ..., ±∞), H_mn(T) ={λ ∈ C: nul(T - λ) = m and nul(T - λ)* = n} (m,n = 0, 1, ..., ∞) Let D° denote the interior of subset D of C and ∂D denote boundary of D. Let σ_N(T) = σ_le(T)＼H∞∞(T)°. If T ∈ L(X) and S ∈ L(Y) are quasisimilar, then for each complex λ nul(T - λ) = nul(S - λ) and nul(T - λ)* = nul(S - λ)*. It follows that H_n(T) = H_n(S) and Hmn(T) = H_mn(S).