Definition 6.2.9: Functional Calculus

Definition 6.2.9 (Functional Calculus).label Let $A$ be a unital Banach algebra and fix $x\in A$. Let $f$ be a holomorphic function in an open neighborhood $U_{f}$ of $\spec{A}{x}$ and $C\suf U_{f}$ be a smooth simple closed curve enclosing $\spec{A}{x}$ with positive orientation. We define

\begin{align*}f(x)\define\frac{1}{2\pi i}\integral{C}{}f(\lam)(\lam-x)\inv d\lam\end{align*}

where the integral is a Bochner intergral in the Banach algebra $A$.

Proof. For each $\p\in A^{*}$, we consider a continuous function $\lam\mto f(\lam)\inner{(\lam-x)\inv}{\p}\in\com$ on the curve $C$. Define a linear functional

\begin{align*}F:A^{*}\to \com&&\p\mto \frac{1}{2\pi i}\integral{C}{}f(\lam)\inner{(\lam-x)\inv}{\p}d\lam\end{align*}

then $F\in A^{**}$ since

\begin{align*}\abs{F(\p)}\leq\frac{l}{2\pi}\norm{\p}{}\SUP{\lam\in\com}\norm{f(\lam)(\lam-x)\inv}{}\end{align*}

where $l$ is the length of the curve $C$. Hence we may write

\begin{align*}\inner{F}{\p}=F(\p)=\frac{1}{2\pi i}\integral{C}{}f(\lam)\inner{(\lam-x)\inv}{\p}d\lam\end{align*}

OTOH, the $A$-valued function $\lam\mto f(\lam)(\lam-x)\inv$ is continuous so the limit

\begin{align*}y\define\frac{1}{2\pi i}\limit{\max\abs{\lam_j-\lam_{j+1}}\to 0}\sums{n}{j=0}f(\lam_{j})(\lam_{j}-x)\inv(\lam_{j}-\lam_{j+1})\end{align*}

exists, where $\curl{\lam_0,...,\lam_n,\lam_{n+1}=\lam_0}$ is a partition of the curve $C$. Then $\inner{y}{\p}=\inner{F}{\p}$ for all $\p\in A^{*}$. Hence $F\in A\suf A^{**}$. By Cauchy’s theorem for contour integrals, $F$ does not depend on the curve $C$. Therefore we denote $F$ by $f(x)$ and write

\begin{align*}f(x)=\frac{1}{2\pi i}\integral{C}{}f(\lam)(\lam-x)\inv d\lam\end{align*}

$\square$

The Eigenspace

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The Eigenspace

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