Boundary Of Spectrum Contained In Approximate Point Spectrum

$σ_{ap}(T) ⊇ ∂σ(T)$, where $∂σ(T)$ is the boundary of $σ(T)$ in the topology of ℂ. part II linear analysis, page 67 theorem 2.2 Proof. Let $λ ∈ ∂ σ(T)$. Then there is a sequence $λ_n ∉ σ(T)$ converging to $λ$. It follows from Corollary 5.2(iii) of the course that \[ \left\|\left(λ_n I-T\right)^{-1}\right\| → ∞   \text { as }   n → ∞ . \] Thus, there is a sequence $\left(x_n\right)$ of unit vectors such that \[ \left\|\left(λ_n I-T\right)^{-1} x_n\right\| → ∞   \text { as }   n → ∞ . \] Set \[ y_n=\frac{\left(λ_n I-T\right)^{-1} x_n}{\left\|\left(λ_n I-T\right)^{-1} x_n\right\|} . \] It is easy to check that $\left(y_n\right)$ is an approximate eigenvector for $λ$.