compact operator

A compact operator is one that 'squeezes' infinite dimensions down toward finite behavior. It takes any bounded set and crushes its image into a set that is almost finite-dimensional — relatively compact, meaning sequences in the image always have convergent subsequences. Compact operators are the closest infinite-dimensional cousins of finite-rank matrices, and they behave almost as well.

Precisely, a linear operator T between normed spaces is compact if the image T(B) of the unit ball B has compact closure (equivalently, every bounded sequence (x_n) has a subsequence with (T x_n) convergent). On a Hilbert or Banach space the compact operators form a closed two-sided ideal in the algebra of bounded operators, and they are exactly the norm-limits of finite-rank operators (for Hilbert spaces and many Banach spaces).

Their spectral theory is exceptionally clean. By the Riesz–Schauder theory, a compact operator on an infinite-dimensional space has spectrum consisting of 0 together with at most countably many nonzero eigenvalues, each of finite multiplicity, with 0 as the only possible accumulation point. The caveat: 0 always lies in the spectrum on an infinite-dimensional space (the identity is never compact), and 0 need not be an eigenvalue.