= Functional calculus
{title2=$f(T)$}
A functional calculus assigns an <linear operator> $f(T)$ to suitable scalar <functions> $f$ and an operator $T$, extending polynomial substitution
$$
p(T)=a_0I+a_1T+\cdots+a_mT^m.
$$
A useful calculus preserves algebraic identities, such as $(fg)(T)=f(T)g(T)$, on its allowed function class. The admissible functions and continuity properties depend on the operator: polynomial, <holomorphic functional calculus>, continuous and measurable spectral calculi are different constructions.
For a diagonal matrix $T=\operatorname{diag}(\lambda_1,\ldots,\lambda_n)$, substitution gives $f(T)=\operatorname{diag}(f(\lambda_1),\ldots,f(\lambda_n))$. A nondiagonal Jordan block shows why values alone need not suffice: if $T=\lambda I+N$ with $N^2=0$, then $p(T)=p(\lambda)I+p'(\lambda)N$. This follows by expanding each power and discarding all terms containing $N^2$.
For bounded <self-adjoint operators>, the <spectral theorem> supplies a calculus for continuous functions on the <spectrum (functional analysis)>, and more generally bounded measurable functions through spectral integration. It defines objects such as positive square roots without choosing a basis of eigenvectors.
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