The Riesz Functional Calculus

Let $\mathcal{A} = C(X)$, $X$ compact (see Example 3.2). If $g \in C(X)$ and $f \in \text{Hol}(g)$, show that $f(g) = f \circ g$.

Let $a$ be a nilpotent element of $\mathcal{A}$. For $f, g$ in $\text{Hol}(a)$, give a necessary and sufficient condition on $f$ and $g$ that $f(a) = g(a)$.

Let $d \ge 1$ and let $A \in M_d(\mathbb {C})$. Give a necessary and sufficient condition on $f$ in $\text{Hol}(A)$ such that $f(A) = 0$.

If $\mathcal{A}$ is a Banach algebra with identity, $a \in \mathcal{A}$, $f \in \text{Hol}(a)$, and $g$ is analytic in a neighborhood of $f(\sigma (a))$, then $g \circ f \in \text{Hol}(a)$ and $g(f(a)) = g \circ f(a)$.

If $\mathcal{X}$ is a Banach space, $A \in \mathcal{B}(\mathcal{X})$, and $\mathcal{M} \le \mathcal{X}$ such that $(A - \alpha )^{-1} \mathcal{M} \subseteq \mathcal{M}$ for all $\alpha$ in $\rho (A)$, show that $f(A) \mathcal{M} \subseteq \mathcal{M}$ whenever $f \in \text{Hol}(A)$.

If $\mathcal{X}$ is a Banach space, $A \in \mathcal{B}(\mathcal{X})$, and $f \in \text{Hol}(A)$, show that $f(A)^* = f(A^*)$. (See (6.1) below.)

If $\mathcal{H}$ is a Hilbert space, $A \in \mathcal{B}(\mathcal{H})$, and $f \in \text{Hol}(A)$, show that $f(A)^* = \tilde{f}(A^*)$, where $\tilde{f}(z) = \overline{f(\bar{z})}$. (See (6.1) below.)

If $\mathcal{H}$ is a Hilbert space, $A$ is a normal operator on $\mathcal{H}$, and $f \in \text{Hol}(A)$, show that $f(A)$ is normal.

Let $\mathcal{X}$ be a Banach space and let $A \in \mathcal{B}(\mathcal{X})$. Show that if $\sigma (A) = F_1 \cup F_2$ where $F_1, F_2$ are disjoint closed subsets of $\mathbb {C}$, then there are topologically complementary subspaces $\mathcal{X}_1, \mathcal{X}_2$ of $\mathcal{X}$ such that

Let $A \in M_d(\mathbb {C})$, $\sigma (A) = \{ \alpha_1, \ldots , \alpha_n\}$, where $\alpha_i \ne \alpha_j$ for $i \ne j$. Show that for $1 \le j \le n$ there is a matrix $A_j$ in $M_{d_j}(\mathbb {C})$ such that $\sigma (A_j) = \{ \alpha_j\}$ and $A$ is similar to $A_1 \oplus \cdots \oplus A_n$.

If $\mathcal{A}$ is a Banach algebra, $I$ is an ideal of $\mathcal{A}$ (not necessarily closed), $a \in I$, and $f \in \text{Hol}(a)$ such that $f(0) = 0$, show that $f(a) \in I$.