Let $\hat{A}$ be a linear operator with know eigenvectors and values $\hat{A}\vec{v}_i = v_i\vec{v}_i$. Is it true and under what condition one can say that $f(\hat{A})\vec{v}_i = f(v_i)\vec{v}_i$? It seems that "sufficient" smoothness of function $f$ is almost certainly required. Is there conditions that dimension of the linear space must be finite? Any other important conditions?



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