Assess graduate-level ability to analyze spectral properties and pseudospectra of non-normal operators on Hilbert and Banach spaces: include precise definitions (spectrum, resolvent, numerical range, ε-pseudospectrum), derive resolvent-norm and pseudospectral bounds, compare spectrum vs pseudospectrum through finite- and infinite-dimensional examples (Jordan blocks, Toeplitz matrices, differential operators), construct proofs or counterexamples demonstrating spectral instability and spectral pollution, and explain implications for semigroup stability and numerical approximation methods.