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} - 2. Adjoint operator - <Tx, y = <x, T y defines T - For matrices: T = conjugate transpose - 3. Spectral Theory - Spectrum: sigma(T) = {lambda : T - lambda I not invertible} - Self-adjoint: spectrum is real - 4. Compact operators - T compact if T(bounded set) has compact closure - Approximable by finite-rank operators - 5. Spectral Theorem - Self-adjoint compact:…