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Let $ A$ be a complex unital Banach algebra. An element $a \in A$ is said tobe Hermitian, if $ \| \exp (ita) \| =1$ for all $t\in R$. In the case of thealgebra of bounded linear operators in a Hilbert space this Hermitian propertyagrees with the ordinary selfadjointness. If $a \in A$ is Hermitian, then$|a|=||a||$, where $|a|$ denotes the spectral radius of $a$. A function $F:R\to \C$ is called the universal symbol if $ \|| F(a)||= |F(a)|$\ for each $ A$and all Hermitian $a\in A$. We characterize universal symbols in terms ofpositive definite functions.

On functional calculus for Hermitian elements of Banach algebras: the norm and spectral radius