 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/ii/paper-4/22h/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

σ (T) = {λ \in C : T - λ I has no bounded two-sided inverse}, ρ (T) = C ∖ σ (T) .

The bounded inverse theorem says that a bounded bijection between Banach spaces automatically has a bounded inverse, so failure of bijectivity suffices here. The point spectrum is

σ_{p} (T) = {λ \in C : ker (T - λ I) \neq = {0}},

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