Assess graduate-level understanding of compact operators on Hilbert spaces: definitions and equivalent characterizations (compactness, sequential compactness, finite-rank approximations), spectral structure (point spectrum, eigenvalue accumulation only at 0, multiplicity), singular value decomposition for compact (self-adjoint and general) operators, and illustrative examples (integral operators, unilateral shift, finite-rank perturbations). Include short proofs of key properties, calculations of spectra or singular values for given operators, and brief explanations of consequences such as the Fredholm alternative.