As the trace of a matrix equals to the sum of all eigenvalues, do we have an analogous result that the trace of an operator (if it is well-defined) equals to an integral on the spectrum space?
2026-04-01 20:01:08.1775073668
Relations between Trace and Spectrum of an Operator
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It seems to me that you're after Lidskii's theorem. It states that if $H$ is a separable Hilbert space, if $A\colon H\longrightarrow H$ is an operator of trace class, and if $\{\lambda_1,\lambda_2,\ldots\}$ are the eigenvalues of $A$ (enumerated with algebraic multiplicities taken into account), then$$\sum_{n=1}^\infty\lambda_n=\operatorname{Tr}(A).$$Note that the previous sum may well be just a finite sum.