operator-theory

# Operator Theory ## When to Use Use this skill when working on operator-theory problems in functional analysis. ## Decision Tree 1. **Bounded operator verification** - ||Tx|| <= M||x|| for some M - Operator norm: ||T|| = sup{||Tx|| : ||x|| = 1} - `z3_solve.py prove "operator_bounded"` 2. **Adjoint operator** - <Tx, y> = <x, T*y> defines T* - For matrices: T* = conjugate transpose - `sympy_compute.py simplify "<Tx, y> - <x, T*y>"` 3. **Spectral Theory** - Spectrum: sigma(T) = {lambda : T - lambda*I not invertible} - Self-adjoint: spectrum is real - `z3_solve.py prove "self_adjoint_real_spectrum"` 4. **Compact operators** - T compact if T(bounded set) has compact closure - Approximable by finite-rank operators - `sympy_compute.py limit "||T - T_n||" --var n` 5. **Spectral Theorem** - Self-adjoint compact: T = sum(lambda_n * P_n) - eigenvalues -> 0, eigenvectors form orthonormal basis ## Tool Commands ### Z3_Bounded_Operator ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "norm(Tx) <= M*norm(x)" ``` ### Sympy_Adjoint ```bash uv run python -m runtime.harness scripts/sympy_compute.py simplify "<Tx, y> - <x, T_star_y>" ``` ### Z3_Spectral ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "self_adjoint implies real_spectrum" ``` ### Sympy_Compact ```bash uv run python -m runtime.harness scripts/sympy_compute.py limit "norm(T - T_n)" --var n --at oo ``` ## Key Techniques *From indexed text...