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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$\begingroup$ See Functional calculus wiki page and the "see also"s. $\endgroup$– user10354138Jul 11, 2021 at 9:34
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$\begingroup$ @user10354138 Text on holomorphic functional calculus seems relevant. $\endgroup$– i_prob_should_know_thisJul 11, 2021 at 9:42
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