Compact operators and the Fredholm alternative

A compact linear operator maps every bounded sequence to an output sequence with a convergent subsequence. In normed spaces this is equivalent to relative compactness of the image of the unit ball. Every compact linear operator is bounded, and every finite-rank bounded operator is compact. On an infinite-dimensional Hilbert space, the identity is not compact: an orthonormal sequence stays a fixed distance apart. Calling its unit ball closed and bounded does not make that ball compact.

The diagonal map K on square-summable sequences defined by K e_n=e_n/n is compact. Truncate it after N coordinates to obtain a finite-rank map K_N. The operator-norm error is 1/(N+1), so K is a norm limit of compact operators and is itself compact. This example gives an explicit approximation certificate. Pointwise convergence of finite-rank maps alone would be weaker and would not prove compactness of the limit; projections onto the first N coordinates converge strongly to the noncompact identity.

For a compact operator on a complex infinite-dimensional Banach space, every nonzero spectral value is an eigenvalue with finite algebraic multiplicity. Such eigenvalues can accumulate only at zero. Zero is always in the spectrum on an infinite-dimensional space, since a compact invertible operator would make the identity compact after composition with its bounded inverse. Yet zero need not be an eigenvalue: the diagonal K above has zero kernel, although its inverse on its range is unbounded.

This distinction is easy to miss when extrapolating from matrices. Each nonzero coordinate multiplier 1/n can be inverted separately, but the inverse multipliers n have no uniform bound. The exceptional point zero records a stability failure without a nonzero vector killed by K. Compact self-adjoint operators admit a particularly strong orthonormal eigenvector description; general compact operators can have Jordan structure and do not inherit orthogonality merely because they are compact.

For compact K on a Hilbert space, I−K has closed range and finite-dimensional kernel and cokernel with equal dimensions. It is invertible exactly when its kernel is zero. At an exceptional eigenvalue, the equation (I−K)x=b is solvable precisely when b is perpendicular to the kernel of I−K*. If it is solvable, solutions differ by vectors in the kernel. This is the Fredholm alternative in a form that separates existence, uniqueness, and compatibility.

The statement concerns an identity minus a compact operator, not an arbitrary compact equation Kx=b and not every bounded linear map. In applications, a differential equation may first be transformed using a Green operator into this form. One must justify that transformation, its boundary conditions, and compactness in the selected function spaces. A finite discretization always has matrix rank conditions, but those alone do not certify the corresponding infinite-dimensional Fredholm hypotheses.