Spectral-Theory

Oct 17, 2021 Functional Analysis 68 min read

Functional Analysis (9): Unbounded Operators — When Boundedness Fails

Closed operators, the distinction between symmetric and self-adjoint, deficiency indices, Friedrichs extension, the spectral theorem for unbounded self-adjoint operators, and Stone's theorem.

Oct 15, 2021 Functional Analysis 68 min read

Functional Analysis (8): Spectral Theory — Decomposing Operators

The spectrum generalizes eigenvalues to infinite dimensions — the spectral theorem for bounded self-adjoint operators and continuous functional calculus give us a complete decomposition.

Oct 13, 2021 Functional Analysis 62 min read

Functional Analysis (7): Compact Operators — The Bridge to Finite Dimensions

Compact operators are limits of finite-rank operators and inherit much finite-dimensional spectral behavior — the Fredholm alternative and spectral theorem for compact self-adjoint operators.