Can We Obtain the Form of L from Z(u)?

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given the function (or distribution)

\sum_{n=0}^{\infty} f(E_n,u )= Z(u) for 'f' an arbitrary function and E_n a set of eigenvalues of a certain operator f (L) with L self adjoint so all the eigenvalues are real , could we obtain the form of 'L' from Z(u) ??
 
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Define f(E_n,u )
 
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