A trace class operator is a type of compact operator on a Hilbert space that has a well-defined trace, which is the sum of its singular values. These operators are significant in functional analysis because they have nice properties related to the spectrum of operators, particularly regarding the convergence of series and the behavior of eigenvalues. Trace class operators play a vital role in understanding other operator classes, such as Hilbert-Schmidt operators, and their relationships in terms of spectral theory.
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