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### Understanding the Spectral Theorem for (Bounded) Operators

Your $A$ can be written as $\sum \lambda_i P_i,$ where $P_i$ is the projection onto the eigenspace of $\lambda_i.$ A sum is a finite version of an integral.
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### Is a holomorphic Function of a matrix invertible?

There is this theorem in spectral theory that says (if $g$ is continuous and defined on the spectrum of linear bounded $A$) $$\sigma(g(A)) = g(\sigma(A)).$$ $\sigma$ of course denotes the spectrum. ...
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