for $t=0$, showing that $A$ is [*not*]{} invertible in $\mathcal{M}_n(C([0,1])$.

Finding the maxima and minima over $[0,1]$ of the continuous functions $\lambda_{-}(t)$ and $\lambda_{+}(t)$ we easily find that the spectrum of $A(t)$ is given by $$\sigma(A(t)) = [-0.77664, 0]\cup [2, 3.86280],$$ where the results have been rounded to five decimal digits. Thus, in this simple example $\|\hat A\|_2=4.03586$ gives a pretty good upper bound for the true value $\|A\| = 3.86280$.

The holomorphic functional calculus
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The standard way to define the notion of an analytic function $f(a)$ of an element $a$ of a $C^*$-algebra ${\cal A}_0$ is via contour integration. In particular, we can use this approach to define functions of a matrix $A$ with elements in ${\cal A}_0$.

Let $f(z)$ be a complex function which is analytic in a open neighborhood $U$ of $\sigma(a)$. Since $\sigma(a)$ is compact, we can always find a finite collection $\Gamma=\cup_{j=1}^\ell\gamma_j$ of smooth simple closed curves whose interior parts contain $\sigma(a)$ and entirely contained in $U$. The curves $\gamma_j$ are assumed to be oriented counterclockwise.

Then $f(a)\in\mathcal{A}_0$ can be defined as $$f(a)=\frac{1}{2\pi {\rm i}}\int_\Gamma f(z)(zI-a)^{-1} dz,\label{integral_def}$$ where the line integral of a Banach-space-valued function $g(z)$ defined on a smooth curve $\gamma:t\mapsto z(t)$ for 