Zeta function (operator)

The of a mathematical operator ${\displaystyle {\mathcal {O}}}$ is a function defined as

for those values of s where this expression exists, and as an analytic continuation of this function for other values of s. Here "tr" denotes a functional trace.

The zeta function may also be expressible as a spectral zeta function in terms of the eigenvalues ${\displaystyle \lambda _{i}}$ of the operator ${\displaystyle {\mathcal {O}}}$ by

It is used in giving a rigorous definition to the functional determinant of an operator, which is given by

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