Compact Operators in Spectral Theory

graph TD D1["D1 Compact operator K\nBounded sets → relatively compact"] D2["D2 Hilbert-Schmidt operator\n∑ ||K e_n||² < ∞ for ONS"] D3["D3 Trace class\n|K| = (K*K)^{1/2} has finite trace"] D4["D4 Fredholm index\nindex T = dim ker T − dim coker T"] T1["T1 Riesz-Schauder\nσ(K) discrete, 0 only limit point"] T2["T2 Spectral theorem for compact\nK = ∑ λ_n ⟨·,e_n⟩ e_n, λ_n → 0"] T3["T3 Fredholm alternative\n(I − K)u = f solvable ⇔ f ⊥ ker(I−K*)"] T4["T4 Weyl theorem\nT compact ⇒ σ_ess(T+K) = σ_ess(T)"] T5["T5 Min-max principle\nλ_n = min max ⟨Tx,x⟩ / ||x||²"] D1 --> D2 D2 --> D3 D1 --> D4 D1 --> T1 D1 --> T2 D1 --> T3 D1 --> T4 D2 --> T2 D4 --> T3 T2 --> T5 T1 --> T2 classDef definition fill:#3498db,color:#fff,stroke:#2980b9 classDef theorem fill:#1abc9c,color:#fff,stroke:#16a085 class D1,D2,D3,D4 definition class T1,T2,T3,T4,T5 theorem