Let $A$ be a unital Banach algebra over complex numbers and call elements of $A$ "vectors". Let $\Omega$ be an open set in $\mathbb{C}$ and $H(\Omega)$ the space of analytic functions on $\Omega$. There are two kinds of holomorphic calculus we can do on $A$.
Vector-valued Analytic Functions
A vector-valued function $ F: \Omega \to A $ is called analytic if the limit
$$ F'(z) = \lim_{h \to 0} \frac{F(z+h)-F(z)}{h} $$
exists for all $ z \in \Omega $, where the limit is taken w.r.t. the norm on $A$.
A basic result says that $F$ is analytic if and only if $l \circ F: \Omega \to \mathbb{C} $ is analytic in the classical complex analytic sense for every bounded functional $l \in A^*$.
Many theorems in classical complex analysis can be extended to an vector-valued analytic function $F$. For example, we have the Cauchy integral formula:
$$ F(z) = \frac{1}{2 \pi i} \int_\Gamma \frac{F(\zeta)}{\zeta - z} d\zeta $$
where $\Gamma$ is a simple closed contour in $\Omega$ that encloses $z$. The integral is the Bochner integral, which is defined as the norm limit of Riemann partial sums.
Holomorphic Functional Calculus
Let $a \in A$ be a vector with spectrum $\sigma(a)$ contained in $\Omega$. Any classical analytic function $ f: \Omega \to \mathbb{C} $ gives rise to a vector $f(a) \in A$, which is defined to be the Bochner integral:
$$ f(a) = \frac{1}{2 \pi i} \int_\Gamma \frac{f(\zeta)}{\zeta - a} d\zeta $$
For simplicity let's assume the resolvent set $\Omega - \sigma(a)$ is connected, in which case $\Gamma$ can be any simple closed contour in $\Omega$ that encloses all of $\sigma(a)$ (otherwise $\Gamma$ should be a disjoint union of curves that are located and oriented appropriately).
The holomorphic functional calculus defines a continuous algebra homomorphism $ \Phi_a: H(\Omega) \to A $ by $ \Phi_a(f) = f(a) $ for $f \in H(\Omega)$, which satisfies the spectral mapping theorem:
$$ \sigma(f(a)) = f(\sigma(a)) $$
Some observations
Both kinds of calculus output vectors. The difference is, in the first case, we first fix a vector-valued function $ F: \Omega \to A $, then each complex number $ z \in \Omega $ gives a vector $F(z) \in A$. In the second case, we first fix a vector $a \in A$, then each analytic function $ f \in H(\Omega) $ gives another vector $f(a) \in A$.
Both kinds of calculus have Cauchy integral formulae. The difference is, in the integrand of the first integral, the numerator $F(\zeta)$ is vector while the denominator $\zeta - z$ is number. In the integrand of the second integral, the numerator $f(\zeta)$ is number while the denominator $\zeta - a$ is vector.
Holomorphic functional calculus has the spectral mapping theorem $\sigma(f(a)) = f(\sigma(a))$, but vector-valued analytic functions don't have a similar result for the spectrum $\sigma(F(z))$.
Questions
- Based on observations 1 and 2, I feel that there is sort of duality going on between the two kinds of calculus. Are there some established results on this could-be duality?
- As said in observation 3, there is no spectral mapping theorem for a vector-valued analytic function $ F: \Omega \to A $. Let $\sigma(F)$ be the union of all spectra $ \sigma(F) = \bigcup_{z \in \Omega} \sigma(F(z)) $. What do we know about $\sigma(F)$? Is it the analytic image of a closed set? Is it even closed?
- More about the duality. Let $\Sigma \subseteq \mathbb{C}$ be an open set that contains the closure of $\sigma(F)$ (in case $\sigma(F)$ is not closed). For each $z \in \Omega$ and each analytic function $f \in H(\Sigma)$, we get a vector $f(F(z)) \in A$. Thus, we have a family of maps
$$ \begin{align*} F_f: \Omega &\to A \\ z &\mapsto f(F(z)) \end{align*} \quad \text{for each} \quad f \in H(\Sigma) $$
Is each $F_f$ analytic? How is it related to the orignal function $ F: \Omega \to A $? Do we have some version of Montel's theorem for the family $ \{ F_f \}_{f \in H(\Sigma)} $?
