σ(T) spectrum, ρ(T) resolvent set. R_λ=(T−λI)⁻¹. Spectral radius r(T)=lim ‖Tⁿ‖^{1/n}.
graph TD
D1["Def: σT spectrum\nT−λ not invertible"]
D2["Def: Resolvent ρT\nℂ∖σT"]
D3["Def: R_λ = T−λ⁻¹\nresolvent op"]
D4["Def: Spectral radius\nrT = sup|λ|"]
T1["Thm: σT closed\nnonempty if dim<∞"]
T2["Thm: rT = lim ‖Tⁿ‖¹/ⁿ"]
T3["Thm: Gelfand\nrT = sup spec"]
T4["Thm: Resolvent analytic\non ρT"]
T5["Thm: Neumann series\n‖T‖<1⇒(I−T)⁻¹"]
D1 --> T1
D2 --> D3
D3 --> T4
D4 --> T2
T2 --> T3
T4 --> T5
classDef definition fill:#b197fc,color:#fff
classDef theorem fill:#51cf66,color:#fff
class D1,D2,D3,D4 definition
class T1,T2,T3,T4,T5 theorem
Color Scheme
Violet Definitions
Green Theorems
Process Statistics
- Nodes: 14
- Edges: 11
Frontier: math.SP