=\frac{1}{2\pi {\rm i}}\int_\Gamma f(z)(zI-A)^{-1} dz,\label{integral_def_A}$$ with the obvious meaning of $\Gamma$.

Bounds for the Hermitian case {#herm}
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In this paper we will be concerned mostly with banded and sparse matrices. A matrix $A\in \mathcal{A}$ is [*banded with bandwidth $m$*]{} if $a_{ij}$ is the zero element of $\mathcal{A}_0$ whenever $|i-j|>m$. Banded matrices over a $C^*$-algebra arise in several contexts; see, e.g., [@Berezansky71; @Berezansky98; @CT06; @Osipov] and references therein.

Let $A\in \mathcal{A}$ be a banded Hermitian matrix with bandwidth $m$. In this section we provide exponentially decaying bounds on the norm of the entries $[f(A)]_{ij}$ ($1\le i,j\le n$) of $f(A)$, where $f$ is analytic on a neighborhood of $\sigma(A)$, and we discuss the important case where the bounds do not depend on the order $n$ of the matrix. The results in this section extend to the $C^*$-algebra setting analogous results for matrices over $\mathbb{R}$ or $\mathbb{C}$ found in [@BG99; @BR07] and [@BBR13]. Functions of non-Hermitian (and non-normal) matrices are studied in the following sections.

Hermitian matrices have a real spectrum. If $\sigma(A)\subseteq
[\alpha, \beta]\subset \mathbb{R}$ then, by replacing $A$ (if necessary) with the shifted and scaled matrix $\frac{2}{\beta - \alpha}A - 
\frac{\beta + \alpha}{\beta - \alpha} I_n$, we can assume that 