Singular values of a compact operator

Summary

The singular values of a compact operator $T$ between Hilbert spaces are the eigenvalues of the positive operator $\abs{T} = (T^* T)^{1/2}$, listed in decreasing order. They extend the singular values of matrices to infinite dimensions and form a sequence that tends to $0$. They measure how well $T$ can be approximated by finite-rank operators. Their rate of decay defines the Schatten classes, among them the trace class and the Hilbert–Schmidt operators.